All Questions
36
questions
6
votes
3
answers
788
views
Where is the mistake in the argument in favor of the (erroneous) claim "every Dedekind cut is a rational cut"?
A cut is a set $C$ such that:
(a) $C\subseteq \mathbb Q $
(b) $C \neq \emptyset $
(c) $C \neq \mathbb {Q} $
(d) for all $a, c \in \mathbb Q $ , if $c\in C$ and $a\lt c$ , then $a\in C $
(e) for all $c\...
- 1,796
2
votes
1
answer
97
views
How do I write this Theorem with quantifiers?
Here is the theorem from Steven Abbot's Understanding Analysis.
Theorem. Two real numbers $a$ and $b$ are equal if and only if for every real number $\epsilon > 0$ it follows that $|a - b| < \...
- 139
3
votes
1
answer
90
views
Proposition 5.4.9. Analysis I - Terence Tao.
Proposition 5.4.9 (The non-negative reals are closed). Let $a_1, a_2, a_3, \ldots $ be a Cauchy sequence of non-negative rational numbers. Then $\text{LIM}_{n \to \infty}a_n$ is a non-negative real ...
- 1,430
4
votes
1
answer
273
views
The most explicit way of partitioning the reals into two dense subsets with positive measure
In @UmbertoP's response to the question, "Partition of real numbers into dense subsets of positive measure," the answer is understandable to a advanced undergraduate; however, I have ...
- 1
0
votes
0
answers
45
views
Is given statement indeed true?
I saw saw the statement I am trying to prove in "a proof of lub property" in a Mathematics Stack Exchange post. The statement I am trying to prove is :
$∃\ b ∈ \mathbb {R},(b < a\
\text { ...
- 556
0
votes
0
answers
29
views
Confused on where it says that the inverse is unique on the Property of Existence of a multiplicative inverse [duplicate]
For every number $ a \ne 0 $, there is a number $ a ^{-1} $ such that
$$ a . a^{-1} = a^{-1} . a = 1. $$
This line is from Calculus by Michael Spivak.
I wanted to prove $ (ab)^{-1} = a^{-1}b^{-1} $ ...
0
votes
0
answers
53
views
Can and can’ts of proving inequalities
I was doing the following question and wonder if my proof below is valid
Prove that $\frac{x+z}{y+z} > \frac{x}{y} \to x<y$ if and only if $x<y$
To prove that $\frac{x+z}{y+z} > \frac{x}{...
0
votes
0
answers
31
views
Suppose $(a_n)$ is a sequence of pos. real numbers such that $\lim_{n \to \infty}\frac{a_{n+1}}{a_n} = 0$. Prove that $\lim_{n \to \infty}{a_n} = 0$. [duplicate]
Suppose $(a_n)$ is a sequence of positive real numbers such that
$\lim_{n \to \infty}\frac{a_{n+1}}{a_n} = 0$
Prove that $\lim_{n \to \infty}{a_n} = 0$. (Warning: Be careful not to assume that
$(a_n)$ ...
1
vote
2
answers
93
views
Proving an expression for the limit of a certain sequence [closed]
If $(a_{n})_{n=1}^{\infty}$ is a sequence converging to $L$, with $a_n \geq 0$ for all $n$, how can I prove that $L \geq 0$ and that $(\sqrt{a_n})_{n=1}^{\infty}$ converges to $\sqrt{L}$.
I was under ...
1
vote
1
answer
73
views
Confusion about q-ary system in Zorich's Mathematical analysis
For $p \in \Bbb Z $ there is a q-ary system described which assigns to each real number x in the base q a sequence of {$\alpha_n$} such that $\sum_{i=0}^n \alpha_{p-i}\cdot q^{p-i} \le x \lt q^{p-n} +...
- 41
2
votes
1
answer
327
views
Rudin Principles of Mathematical Analysis Theorem 6.10
Rudin is using the Riemann-Steljis integral and its assumed $\alpha$ is monotonic. Throughout this chapter Rudin has written $M_{i} = \sup f(x)$ where $x_{i-1} \leq x \leq x_{i}$ and $m_{i} = \inf f(x)...
1
vote
1
answer
85
views
Is this lemma correct?
The lemma is stated as an alternative definition for a least upper bound
$$\mathrm{sup} A - \epsilon < a$$
Let $A$ be a set where $A \subseteq \mathbb{R}$ and $\epsilon > 0$, such that for every ...
user956717
0
votes
1
answer
78
views
Clarification for the proof of uncountability of real numbers
The following is from the Understanding Analysis 2nd ed., Stephen Abbot, page 28
If we let $x_1 = f(1), x_2 = f(2)$ and so on, then our assumption that $f$ is onto means that we can write $\mathbb{R} ...
- 1,010
1
vote
1
answer
367
views
Showing that a rational number exists between any two reals
The following is from Real Analysis by N. L. Carothers.
Theorem: If $a$ and $b$ are real numbers with $a < b$, then there is a rational number $r$ $\in \mathbb{Q}$ with $a < r < b$.
Proof: ...
- 1,010
1
vote
0
answers
117
views
Guidance on a proof regarding the uniqueness/existence of the nth root of x.
So in a previous proof I had to show that given $y\in\mathbb{R},n\in\mathbb{N}$ and $\epsilon >0$, show that for some $\delta>0$, if $u\in\mathbb{R}$ and $|u-y|<\delta$ then $|u^n-y^n|<\...
- 3,431