In my previous response to this
topic, I've given a proof of the
positivity using probabilities and integrals. Let me give a different proof
now, which still uses a bit of analysis (limits), but otherwise is completely
combinatorial; its main ingredients are the principle of inclusion and
exclusion and the "tensor power trick" (in a really simple incarnation).
Strengthening the inequality I
We fix a positive integer $n$. We let $\left[ n\right] $ denote the set
$\left\{ 1,2,\ldots,n\right\} $.
We must prove the following fact (I'm copying the numbering from my previous response):
Theorem 2. Let $A_{1},A_{2},\ldots,A_{n}$ be $n$ finite nonempty sets.
Then,
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}A_{i}\right\vert }>0.
\end{equation}
I shall prove the following stronger inequality:
Theorem 3. Let $A_{1},A_{2},\ldots,A_{n}$ be $n$ finite nonempty sets.
Then,
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}A_{i}\right\vert }\geq\dfrac{1}{\left|A_1\right|}.
\end{equation}
Clearly, Theorem 3 implies Theorem 2 (because $\dfrac{1}{\left|A_1\right|}>0$).
Two combinatorial lemmas
We will use the Principle of Inclusion and Exclusion in the following form
(Theorem 3.42 in Darij Grinberg, Notes on the combinatorial fundamentals of
algebra, version of 10 January 2019):
Theorem 4 (Principle of Inclusion and Exclusion). Let $G$ be a finite set.
For each $i\in G$, let $A_i$ be a finite set. Then,
\begin{equation}
\left\vert \bigcup_{i\in G}A_{i}\right\vert
=
\sum_{\substack{I\subseteq G;\\I\neq\varnothing}}
\left( -1\right) ^{\left\vert I\right\vert -1}
\left\vert \bigcap_{i\in I}A_{i}\right\vert .
\end{equation}
But I don't know how to derive Theorem 3 from Theorem 4 directly. Instead, I
will take a detour through the following lemmas:
Lemma 5. Let $Q$ be a finite totally ordered set. Let $J$ be a subset of
$Q$. Let $r\in J$. Let $S$ be the set of all permutations of $Q$. Then,
\begin{equation}
\left\vert \left\{ \sigma\in S\ \mid\ \sigma\left( r\right) >\sigma\left(
j\right) \text{ for all }j\in J\setminus\left\{ r\right\} \right\}
\right\vert =\dfrac{\left\vert S\right\vert }{\left\vert J\right\vert }.
\end{equation}
Lemma 5 is more or less obvious if you have the right intuition: For a
uniformly random permutation $\sigma\in S$, every element $r$ of $J$ is
equally likely to be $\operatorname{argmax}\left\{ \sigma\left( j\right)
\ \mid\ j\in J\right\} $ (which is tantamount to satisfying $\sigma\left(
r\right) >\sigma\left( j\right) $ for all $j\in J\setminus\left\{
r\right\} $); thus, the probability for $r$ to be $\operatorname{argmax}
\left\{ \sigma\left( j\right) \ \mid\ j\in J\right\} $ is $\dfrac
{1}{\left\vert J\right\vert }$. Essentially, this is saying that the
symmetric group $S$ is, well, symmetric. If you find this convincing,
scroll forward to Lemma 6.
But just in case, let me now give a fully rigorous combinatorial
proof for the sake of completeness.
Proof of Lemma 5. The set $J$ is nonempty (since $r\in J$). Thus, for each
permutation $\sigma\in S$, the set $\sigma\left( J\right) $ is nonempty, and
therefore has a largest element. In other words, for each permutation
$\sigma\in S$, there exists a $j\in J$ for which $\sigma\left( j\right) $ is
maximum (among all $j\in J$). This $j$ is unique (since $\sigma$ is a
permutation, and thus never takes the same value twice). We denote this $j$ by
$m\left( \sigma\right) $. Thus, we have defined a map $m:S\rightarrow J$
that sends each permutation $\sigma\in S$ to $m\left( \sigma\right) \in J$.
Now, let us consider the fibers $m^{-1}\left( \left\{ j\right\} \right) $
of this map $m$ for $j\in J$. Fix two elements $u$ and $v$ of $J$. We claim
that $\left\vert m^{-1}\left( \left\{ u\right\} \right) \right\vert
=\left\vert m^{-1}\left( \left\{ v\right\} \right) \right\vert $.
Indeed, there is clearly a permutation $\gamma\in S$ of $Q$ that satisfies
$\gamma\left( v\right) =u$ and $\gamma\left( J\right) =J$. (Indeed, such a
$\gamma$ can be constructed as follows: Pick some permutation of $J$ that
sends $v$ to $u$; then, extend this permutation to the whole $Q$ in an
arbitrary way.) Consider such a $\gamma$.
Now, let $\tau\in m^{-1}\left( \left\{ u\right\} \right) $. Thus, $\tau\in
S$ is a permutation such that $m\left( \tau\right) =u$. From $m\left(
\tau\right) =u$, we conclude that $\tau\left( u\right) $ is the maximum
value among all the $\tau\left( j\right) $ with $j\in J$ (by the definition
of $m$). In other words, $\tau\left( u\right) =\max\left\{ \tau\left(
j\right) \ \mid\ j\in J\right\} $. Since $\left\{ \tau\left( j\right)
\ \mid\ j\in J\right\} =\tau\left( J\right) $, this rewrites as
$\tau\left( u\right) =\max\left( \tau\left( J\right) \right) $. Now,
\begin{align*}
\left( \tau\circ\gamma\right) \left( v\right) & =\tau\left(
\underbrace{\gamma\left( v\right) }_{=u}\right) =\tau\left( u\right)
=\max\left( \tau\left( \underbrace{J}_{=\gamma\left( J\right) }\right)
\right) =\max\left( \underbrace{\tau\left( \gamma\left( J\right) \right)
}_{\substack{=\left( \tau\circ\gamma\right) \left( J\right) \\=\left\{
\left( \tau\circ\gamma\right) \left( j\right) \ \mid\ j\in J\right\}
}}\right) \\
& =\max\left\{ \left( \tau\circ\gamma\right) \left( j\right)
\ \mid\ j\in J\right\} .
\end{align*}
In other words, $\left( \tau\circ\gamma\right) \left( v\right) $ is the
maximum value among all the $\left( \tau\circ\gamma\right) \left( j\right)
$ with $j\in J$. In other words, $m\left( \tau\circ\gamma\right) =v$ (by the
definition of $m$). Thus, the permutation $\tau\circ\gamma\in S$ belongs to
the fiber $m^{-1}\left( \left\{ v\right\} \right) $. In other words,
$\tau\circ\gamma\in m^{-1}\left( \left\{ v\right\} \right) $.
Now, forget that we fixed $\tau$. We thus have proven that $\tau\circ\gamma\in
m^{-1}\left( \left\{ v\right\} \right) $ for each $\tau\in m^{-1}\left(
\left\{ u\right\} \right) $. Hence, the map
\begin{align*}
m^{-1}\left( \left\{ u\right\} \right) & \rightarrow m^{-1}\left(
\left\{ v\right\} \right) ,\\
\tau & \mapsto\tau\circ\gamma
\end{align*}
is well-defined. This map is furthermore injective (because if two elements
$\tau_{1}$ and $\tau_{2}$ of $m^{-1}\left( \left\{ u\right\} \right) $
satisfy $\tau_{1}\circ\gamma=\tau_{2}\circ\gamma$, then clearly $\tau_{1}
=\tau_{2}$, since $\gamma$ is invertible). Hence, we have constructed an
injective map $m^{-1}\left( \left\{ u\right\} \right) \rightarrow
m^{-1}\left( \left\{ v\right\} \right) $. Hence, $\left\vert m^{-1}\left(
\left\{ u\right\} \right) \right\vert \leq\left\vert m^{-1}\left( \left\{
v\right\} \right) \right\vert $. The same argument (with the roles of $u$
and $v$ interchanged) yields $\left\vert m^{-1}\left( \left\{ v\right\}
\right) \right\vert \leq\left\vert m^{-1}\left( \left\{ u\right\} \right)
\right\vert $. Combining this with $\left\vert m^{-1}\left( \left\{
u\right\} \right) \right\vert \leq\left\vert m^{-1}\left( \left\{
v\right\} \right) \right\vert $, we find $\left\vert m^{-1}\left( \left\{
u\right\} \right) \right\vert =\left\vert m^{-1}\left( \left\{ v\right\}
\right) \right\vert $.
Now, forget that we fixed $u$ and $v$. We thus have shown that $\left\vert
m^{-1}\left( \left\{ u\right\} \right) \right\vert =\left\vert
m^{-1}\left( \left\{ v\right\} \right) \right\vert $ for any two elements
$u$ and $v$ of $J$. Applying this to $v=r$, we thus conclude that
\begin{equation}
\left\vert m^{-1}\left( \left\{ u\right\} \right) \right\vert =\left\vert
m^{-1}\left( \left\{ r\right\} \right) \right\vert \qquad\text{for each
}u\in J. \label{pf.l5.1} \tag{1}
\end{equation}
Now, $\sum_{s\in S}1=\left\vert S\right\vert \cdot1=\left\vert S\right\vert $,
so that
\begin{align*}
\left\vert S\right\vert & =\sum_{s\in S}1=\sum_{u\in J}\underbrace{\sum
_{\substack{s\in S;\\m\left( s\right) =u}}1}_{\substack{=\left\vert \left\{
s\in S\ \mid\ m\left( s\right) =u\right\} \right\vert \cdot1\\=\left\vert
\left\{ s\in S\ \mid\ m\left( s\right) =u\right\} \right\vert }
}=\sum_{u\in J}\left\vert \underbrace{\left\{ s\in S\ \mid\ m\left(
s\right) =u\right\} }_{=m^{-1}\left( \left\{ u\right\} \right)
}\right\vert =\sum_{u\in J}\underbrace{\left\vert m^{-1}\left( \left\{
u\right\} \right) \right\vert }_{\substack{=\left\vert m^{-1}\left(
\left\{ r\right\} \right) \right\vert \\\text{(by \eqref{pf.l5.1})}}}\\
& =\sum_{u\in J}\left\vert m^{-1}\left( \left\{ r\right\} \right)
\right\vert =\left\vert J\right\vert \cdot\left\vert m^{-1}\left( \left\{
r\right\} \right) \right\vert ,
\end{align*}
and thus $\left\vert m^{-1}\left( \left\{ r\right\} \right) \right\vert
=\dfrac{\left\vert S\right\vert }{\left\vert J\right\vert }$.
But if $\sigma\in S$ is an arbitrary permutation, then we have the following
equivalences:
\begin{align*}
& \ \left( m\left( \sigma\right) =r\right) \\
& \Longleftrightarrow\ \left( r\text{ is the unique }j\in J\text{ for which
}\sigma\left( j\right) \text{ is maximum}\right) \\
& \qquad\left( \text{by the definition of }m\left( \sigma\right) \right)
\\
& \Longleftrightarrow\ \left(
\begin{array}
[c]{c}
\sigma\left( r\right) \text{ is the largest value among all }\sigma\left(
j\right) \text{ with }j\in J\text{,}\\
\text{and is only attained for }j=r
\end{array}
\right) \\
& \Longleftrightarrow\ \left( \sigma\left( r\right) >\sigma\left(
j\right) \text{ for all }j\in J\setminus\left\{ r\right\} \right) .
\end{align*}
Hence,
\begin{align*}
\left\{ \sigma\in S\ \mid\ m\left( \sigma\right) =r\right\}
& =\left\{ \sigma\in S\ \mid\ \sigma\left( r\right) >\sigma\left(
j\right) \text{ for all }j\in J\setminus\left\{ r\right\} \right\} .
\end{align*}
Thus,
\begin{align*}
& \left\{ \sigma\in S\ \mid\ \sigma\left( r\right) >\sigma\left(
j\right) \text{ for all }j\in J\setminus\left\{ r\right\} \right\} \\
& =\left\{ \sigma\in S\ \mid\ m\left( \sigma\right) =r\right\}
=m^{-1}\left( \left\{ r\right\} \right) .
\end{align*}
Hence,
\begin{align*}
& \left\vert \left\{ \sigma\in S\ \mid\ \sigma\left( r\right)
>\sigma\left( j\right) \text{ for all }j\in J\setminus\left\{ r\right\}
\right\} \right\vert \\
& =\left\vert m^{-1}\left( \left\{ r\right\} \right) \right\vert
=\dfrac{\left\vert S\right\vert }{\left\vert J\right\vert }.
\end{align*}
This proves Lemma 5. $\blacksquare$
Auxiliary inequalities
Time to get to the interesting parts. The following lemma can be viewed as a
particular case of Theorem 3 (applied to $A_i = B_i \cup\left\{ q\right\} $
for some extra element $q$), but we will rather use it as a stepping stone in
our proof of Theorem 3:
Lemma 6. Let $B_{1},B_{2},\ldots,B_{n}$ be $n$ finite sets. Then,
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}B_{i}\right\vert +1}\geq\dfrac{1}{\left|B_1\right|+1}.
\end{equation}
Proof of Lemma 6. Let
$N=\left\vert \bigcup_{i=1}^{n}B_{i}\right\vert $.
Thus, we can WLOG assume that $\bigcup_{i=1}^{n}B_{i}$ is the set $\left\{
1,2,\ldots,N\right\} $ (indeed, we can achieve this by relabeling the
elements of $\bigcup_{i=1}^{n}B_{i}$). Assume this. Thus, $B_{i}
\subseteq\left\{ 1,2,\ldots,N\right\} $ for all $i\in\left[ n\right] $.
Notice that $0\notin B_{i}$ for all $i\in\left[ n\right] $ (since
$B_{i}\subseteq\left\{ 1,2,\ldots,N\right\} $). Hence, $0\notin\bigcup_{i\in
I}B_{i}$ for any subset $I$ of $\left[ n\right] $.
Let $S$ be the set of all permutations of the $\left( N+1\right) $-element
set $\left\{ 0,1,\ldots,N\right\} $. Clearly, $\left\vert S\right\vert
=\left( N+1\right) !$.
For each $i\in\left[ n\right] $, we define a subset $A_{i}$ of $S$ by
\begin{equation}
A_{i}=\left\{ \sigma\in S\ \mid\ \sigma\left( 0\right) >\sigma\left(
j\right) \text{ for all }j\in B_{i}\right\} .
\end{equation}
For any subset $I$ of $\left[n\right]$, we have
\begin{align*}
\bigcap_{i\in I}A_{i} & =\bigcap_{i\in I}\left\{ \sigma\in S\ \mid
\ \sigma\left( 0\right) >\sigma\left( j\right) \text{ for all }j\in
B_{i}\right\} \\
& \qquad\left( \text{by the definition of }A_{i}\right) \\
& =\left\{ \sigma\in S\ \mid\ \sigma\left( 0\right) >\sigma\left(
j\right) \text{ for all }j\in\underbrace{\bigcup_{i\in I}B_{i}}
_{\substack{=\left( \left\{ 0\right\} \cup\bigcup_{i\in I}B_{i}\right)
\setminus\left\{ 0\right\} \\\text{(since }0\notin\bigcup_{i\in I}
B_{i}\text{)}}}\right\} \\
& =\left\{ \sigma\in S\ \mid\ \sigma\left( 0\right) >\sigma\left(
j\right) \text{ for all }j\in\left( \left\{ 0\right\} \cup\bigcup_{i\in
I}B_{i}\right) \setminus\left\{ 0\right\} \right\}
\end{align*}
and thus
\begin{align*}
\left\vert \bigcap_{i\in I}A_{i}\right\vert & =\left\vert \left\{ \sigma\in
S\ \mid\ \sigma\left( 0\right) >\sigma\left( j\right) \text{ for all }
j\in\left( \left\{ 0\right\} \cup\bigcup_{i\in I}B_{i}\right)
\setminus\left\{ 0\right\} \right\} \right\vert \\
& =\dfrac{\left\vert S\right\vert }{\left\vert \left\{ 0\right\}
\cup\bigcup_{i\in I}B_{i}\right\vert }
\end{align*}
(by Lemma 5, applied to $Q=\left\{ 0,1,\ldots,N\right\} $, $J=\left\{
0\right\} \cup\bigcup_{i\in I}B_{i}$ and $r=0$). Hence, for any subset $I$ of
$\left[ n \right]$, we have
\begin{equation}
\left\vert \bigcap_{i\in I}A_{i}\right\vert =\dfrac{\left\vert S\right\vert
}{\left\vert \left\{ 0\right\} \cup\bigcup_{i\in I}B_{i}\right\vert }
=\dfrac{\left\vert S\right\vert }{\left\vert \bigcup_{i\in I}B_{i}\right\vert
+1} \label{pf.l6.4} \tag{2}
\end{equation}
(since $\left\vert \left\{ 0\right\} \cup\bigcup_{i\in I}B_{i}\right\vert
=\left\vert \bigcup_{i\in I}B_{i}\right\vert +1$ (because $0\notin
\bigcup_{i\in I}B_{i}$)).
Also, $\bigcup_{i\in\left[ n\right] }A_{i}\supseteq A_1
= \bigcap_{i\in\left\{
1\right\} }A_{i}$ and thus
\begin{align*}
\left\vert \bigcup_{i\in\left[ n\right] }A_{i}\right\vert & \geq\left\vert
\bigcap_{i\in \left\{1\right\} }A_{i}\right\vert =\dfrac{\left\vert
S\right\vert }{\left\vert \bigcup_{i\in \left\{1\right\}}B_{i}\right\vert
+1}\\
& \qquad\left( \text{by \eqref{pf.l6.4} (applied to }I=\left\{1\right\}
\text{)}\right) \\
& =\dfrac{\left\vert S\right\vert }{\left|B_1\right|+1}
\end{align*}
(since $\bigcup_{i\in \left\{1\right\} }B_{i}
= B_1$).
Now, Theorem 4 (applied to $G=\left[ n\right] $) yields
\begin{equation}
\left\vert \bigcup_{i\in\left[ n\right] }A_{i}\right\vert =\sum
_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing}}\left(
-1\right) ^{\left\vert I\right\vert -1}\underbrace{\left\vert \bigcap_{i\in
I}A_{i}\right\vert }_{\substack{=\dfrac{\left\vert S\right\vert }{\left\vert
\bigcup_{i\in I}B_{i}\right\vert +1}\\\text{(by \eqref{pf.l6.4})}}
}=\sum_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing}}\left(
-1\right) ^{\left\vert I\right\vert -1}\dfrac{\left\vert S\right\vert
}{\left\vert \bigcup_{i\in I}B_{i}\right\vert +1},
\end{equation}
so that
\begin{equation}
\sum_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing}}\left(
-1\right) ^{\left\vert I\right\vert -1}\dfrac{\left\vert S\right\vert
}{\left\vert \bigcup_{i\in I}B_{i}\right\vert +1}=\left\vert \bigcup
_{i\in\left[ n\right] }A_{i}\right\vert \geq\dfrac{\left\vert S\right\vert
}{\left|B_1\right|+1}.
\end{equation}
Cancelling $\left\vert S\right\vert $ from this inequality (this is allowed
since $\left\vert S\right\vert >0$), we obtain
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}B_{i}\right\vert +1}\geq\dfrac{1}{\left|B_1\right|+1}.
\end{equation}
This proves Lemma 6. $\blacksquare$
Lemma 7. Let $A_{1},A_{2},\ldots,A_{n}$ be $n$ finite sets.
Let $m$ be a positive integer. Then,
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}A_{i}\right\vert +1/m}\geq\dfrac{1}{\left|A_1\right|+1/m}.
\end{equation}
Proof of Lemma 7. For each $i\in\left[ n\right] $, we define a finite set
$B_{i}$ by $B_{i}=A_{i}\times\left\{ 1,2,\ldots,m\right\} $. Then, for each
subset $I$ of $\left[ n\right] $, we have
\begin{equation}
\bigcup_{i\in I}\underbrace{B_{i}}_{=A_{i}\times\left\{ 1,2,\ldots,m\right\}
}=\bigcup_{i\in I}\left( A_{i}\times\left\{ 1,2,\ldots,m\right\} \right)
=\left( \bigcup_{i\in I}A_{i}\right) \times\left\{ 1,2,\ldots,m\right\}
\end{equation}
and thus
\begin{align}
\left\vert \bigcup_{i\in I}B_{i}\right\vert & =\left\vert \left(
\bigcup_{i\in I}A_{i}\right) \times\left\{ 1,2,\ldots,m\right\} \right\vert
=\left\vert \bigcup_{i\in I}A_{i}\right\vert \cdot\underbrace{\left\vert
\left\{ 1,2,\ldots,m\right\} \right\vert }_{=m}\nonumber\\
& =\left\vert \bigcup_{i\in I}A_{i}\right\vert \cdot m.\label{pf.l7.2} \tag{3}
\end{align}
Also, the definition of $B_1$ yields
$B_1 = A_1 \times \left\{1,2,\ldots,m\right\}$, so that
\begin{align}
\left|B_1\right| = \left| A_1 \times \left\{1,2,\ldots,m\right\}\right|
= \left|A_1\right| \cdot \underbrace{\left\vert
\left\{ 1,2,\ldots,m\right\} \right\vert }_{=m}
& = \left|A_1\right| \cdot m .
\label{pf.l7.3} \tag{4}
\end{align}
But Lemma 6 yields
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}B_{i}\right\vert +1}\geq\dfrac{1}{\left|B_1\right|+1}.
\end{equation}
In view of \eqref{pf.l7.2} and \eqref{pf.l7.3}, this rewrites as
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}A_{i}\right\vert \cdot m+1}\geq\dfrac{1}{\left|A_1\right| \cdot m+1}.
\end{equation}
Multiplying both sides of this inequality by $m$, we find
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}A_{i}\right\vert +1/m}\geq\dfrac{1}{\left|A_1\right|+1/m}
\end{equation}
(since $m\cdot\dfrac{1}{\left\vert \bigcup_{i\in I}A_{i}\right\vert \cdot
m+1}=\dfrac{1}{\left\vert \bigcup_{i\in I}A_{i}\right\vert +1/m}$ for each
$I\subseteq\left[ n\right] $, and since $m\cdot\dfrac{1}{\left|A_1\right| \cdot m+1}=\dfrac
{1}{\left|A_1\right|+1/m}$). This proves Lemma 7. $\blacksquare$
We can now prove Theorem 3:
Proof of Theorem 3. Consider a positive integer $m$ going to infinity. Then,
$\lim\limits_{m\rightarrow\infty}\dfrac{1}{q+1/m}=\dfrac{1}{q}$ for every
positive real $q$. Thus,
\begin{align*}
& \lim\limits_{m\rightarrow\infty}\sum\limits_{\substack{I\subseteq\left[
n\right] ;\\I\neq\varnothing}}\left( -1\right) ^{\left\vert I\right\vert
-1}\dfrac{1}{\left\vert \bigcup_{i\in I}A_{i}\right\vert +1/m}\\
& =\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\underbrace{\lim
\limits_{m\rightarrow\infty}\dfrac{1}{\left\vert \bigcup_{i\in I}
A_{i}\right\vert +1/m}}_{=\dfrac{1}{\left\vert \bigcup_{i\in I}A_{i}
\right\vert }}=\sum\limits_{\substack{I\subseteq\left[ n\right]
;\\I\neq\varnothing}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac
{1}{\left\vert \bigcup_{i\in I}A_{i}\right\vert },
\end{align*}
so that
\begin{align*}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}A_{i}\right\vert } & =\lim\limits_{m\rightarrow\infty
}\underbrace{\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq
\varnothing}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac
{1}{\left\vert \bigcup_{i\in I}A_{i}\right\vert +1/m}}_{\substack{\geq
\dfrac{1}{\left|A_1\right|+1/m}\\\text{(by Lemma 7)}}}\\
& \geq\lim\limits_{m\rightarrow\infty}\dfrac{1}{\left|A_1\right|+1/m}=\dfrac{1}{\left|A_1\right|}.
\end{align*}
This proves Theorem 3. $\blacksquare$
Strengthening the inequality II
But wait... We can prove better inequalities. Let us extend Lemma 6 as follows:
Lemma 8. Let $B_{1},B_{2},\ldots,B_{n}$ be $n$ finite sets. Let $G$ be a
subset of $\left[ n\right] $. Then,
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}B_{i}\right\vert +1}\geq\sum\limits_{\substack{I\subseteq
G;\\I\neq\varnothing}}\left( -1\right) ^{\left\vert I\right\vert -1}
\dfrac{1}{\left\vert \bigcup_{i\in I}B_{i}\right\vert +1}.
\end{equation}
Note that Lemma 8 turns into Lemma 6 if you set $G=\left\{ 1\right\} $.
Proof of Lemma 8. Let us use the same notations as in the proof of Lemma 6.
Let us also WLOG assume that $\bigcup_{i=1}^{n}B_{i}$ is the set $\left\{
1,2,\ldots,N\right\} $ (as in the proof of Lemma 6).
From $G\subseteq\left[ n\right] $, we obtain $\bigcup_{i\in G}A_{i}
\subseteq\bigcup_{i\in\left[ n\right] }A_{i}$, and thus
\begin{equation}
\left\vert \bigcup_{i\in G}A_{i}\right\vert \leq\left\vert \bigcup
_{i\in\left[ n\right] }A_{i}\right\vert =\sum_{\substack{I\subseteq\left[
n\right] ;\\I\neq\varnothing}}\left( -1\right) ^{\left\vert I\right\vert
-1}\dfrac{\left\vert S\right\vert }{\left\vert \bigcup_{i\in I}B_{i}
\right\vert +1}
\end{equation}
(this is proven as in the proof of Lemma 6). Thus,
\begin{align*}
& \sum_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing}}\left(
-1\right) ^{\left\vert I\right\vert -1}\dfrac{\left\vert S\right\vert
}{\left\vert \bigcup_{i\in I}B_{i}\right\vert +1}\\
& \geq\left\vert \bigcup_{i\in G}A_{i}\right\vert =\sum_{\substack{I\subseteq
G;\\I\neq\varnothing}}\left( -1\right) ^{\left\vert I\right\vert
-1}\underbrace{\left\vert \bigcap_{i\in I}A_{i}\right\vert }
_{\substack{=\dfrac{\left\vert S\right\vert }{\left\vert \bigcup_{i\in I}
B_{i}\right\vert +1}\\\text{(by \eqref{pf.l6.4})}}}\\
& \qquad\left( \text{by Theorem 4}\right) \\
& =\sum_{\substack{I\subseteq G;\\I\neq\varnothing}}\left( -1\right)
^{\left\vert I\right\vert -1}\dfrac{\left\vert S\right\vert }{\left\vert
\bigcup_{i\in I}B_{i}\right\vert +1}.
\end{align*}
Cancelling $\left\vert S\right\vert $ from this inequality (this is allowed
since $\left\vert S\right\vert >0$), we obtain
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}B_{i}\right\vert +1}\geq\sum\limits_{\substack{I\subseteq
G;\\I\neq\varnothing}}\left( -1\right) ^{\left\vert I\right\vert -1}
\dfrac{1}{\left\vert \bigcup_{i\in I}B_{i}\right\vert +1}.
\end{equation}
This proves Lemma 8. $\blacksquare$
Next, we generalize Lemma 7 as follows:
Lemma 9. Let $A_{1},A_{2},\ldots,A_{n}$ be $n$ finite sets. Let $G$ be a
subset of $\left[ n\right] $. Let $m$ be a positive integer. Then,
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}A_{i}\right\vert +1/m}\geq\sum\limits_{\substack{I\subseteq
G;\\I\neq\varnothing}}\left( -1\right) ^{\left\vert I\right\vert -1}
\dfrac{1}{\left\vert \bigcup_{i\in I}A_{i}\right\vert +1/m}.
\end{equation}
Proof of Lemma 9. Lemma 9 can be derived from Lemma 8 in the same way as
Lemma 7 was derived from Lemma 6. $\blacksquare$
We can now generalize Theorem 3:
Theorem 10. Let $A_{1},A_{2},\ldots,A_{n}$ be $n$ finite nonempty sets.
Let $G$ be a subset of $\left[ n\right] $. Then,
\begin{equation}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}A_{i}\right\vert }\geq\sum\limits_{\substack{I\subseteq
G;\\I\neq\varnothing}}\left( -1\right) ^{\left\vert I\right\vert -1}
\dfrac{1}{\left\vert \bigcup_{i\in I}A_{i}\right\vert }.
\end{equation}
Proof of Theorem 10. Theorem 10 can be derived from Lemma 9 in the same way
as Theorem 3 was derived from Lemma 7. $\blacksquare$
One interesting feature of Theorem 10 is that (unlike our above proof of
Theorem 3) it allows us to determine when equality holds in Theorem 3. Namely, we have the following:
Proposition 11. Equality holds in Theorem 3 if and only if each
$j\in\left[ n\right] $ satisfies $A_{1}\subseteq A_{j}$.
Proof of Proposition 11 (sketched). I leave it to the reader to check that
if each $j\in\left[ n\right] $ satisfies $A_{1}\subseteq A_{j}$, then
equality holds in Theorem 3 (hint: all addends on the left hand side cancel
each other in pairs, except for the addend $\dfrac{1}{\left\vert
A_{1}\right\vert }$). Let me prove the converse:
Assume that equality holds in Theorem 3. We must show that each $j\in\left[
n\right] $ satisfies $A_{1}\subseteq A_{j}$.
Indeed, assume the contrary. Thus, there exists some $j\in\left[ n\right] $
such that $A_{1}\not \subseteq A_{j}$. Consider this $j$. From $A_{1}
\not \subseteq A_{j}$, we conclude that $A_{j}$ is a proper subset of
$A_{1}\cup A_{j}$. Hence, $\left\vert A_{j}\right\vert <\left\vert A_{1}\cup
A_{j}\right\vert $, so that $\dfrac{1}{\left\vert A_{j}\right\vert }>\dfrac
{1}{\left\vert A_{1}\cup A_{j}\right\vert }$.
Also, $A_{1}\neq A_{j}$ (since $A_{1}\not \subseteq A_{j}$), so that $1\neq
j$. Now, Theorem 10 (applied to $G=\left\{ 1,j\right\} $) yields
\begin{align*}
\sum\limits_{\substack{I\subseteq\left[ n\right] ;\\I\neq\varnothing
}}\left( -1\right) ^{\left\vert I\right\vert -1}\dfrac{1}{\left\vert
\bigcup_{i\in I}A_{i}\right\vert } & \geq\sum\limits_{\substack{I\subseteq
\left\{ 1,j\right\} ;\\I\neq\varnothing}}\left( -1\right) ^{\left\vert
I\right\vert -1}\dfrac{1}{\left\vert \bigcup_{i\in I}A_{i}\right\vert }\\
& =\dfrac{1}{\left\vert A_{1}\right\vert }+\underbrace{\dfrac{1}{\left\vert
A_{j}\right\vert }-\dfrac{1}{\left\vert A_{1}\cup A_{j}\right\vert }
}_{\substack{>0\\\text{(since }\dfrac{1}{\left\vert A_{j}\right\vert }
>\dfrac{1}{\left\vert A_{1}\cup A_{j}\right\vert }\text{)}}}\qquad\left(
\text{since }1\neq j\right) \\
& >\dfrac{1}{\left\vert A_{1}\right\vert }.
\end{align*}
This contradicts our assumption that equality holds in Theorem 3. This
contradiction shows that our assumption was wrong. Hence, Proposition 11 is proven. $\blacksquare$