All Questions
38
questions
0
votes
1
answer
22
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How to more formally prove this inequality
This is a simple problem I came up with while doing another problem:
Given: $n < (n + \frac{1}{2}) < y < (n + 1)$
Prove: $|y - n| > |y - (n + 1)|$
So how I proved it was simply using the ...
- 883
1
vote
0
answers
60
views
Exercise 6, Section 2.2 of Hoffman’s Linear Algebra
(a) Prove that the only subspaces of $\Bbb{R}$ are $\Bbb{R}$ and the zero subspace.
(b) Prove that a subspace of $\Bbb{R}^2$ is $\Bbb{R}^2$, or the zero subspace, or consist of all scalar multiples of ...
- 4,249
6
votes
1
answer
202
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Different definitions of the archimedean property
In some textbooks I have seen the archimedean property defined as:
for some positive real $x$, real number $y$, there exists a natural $n$ such that $nx>y$.
In other textbooks the archimedean ...
0
votes
1
answer
266
views
Proving trichotomy and transitivity from the definition of an ordered field
I'm reading a set of notes (and can provide the link if anyone is interested) which attempt to build up the properties of an ordered field. After defining a field based on its axioms, it defines an ...
- 1,418
0
votes
0
answers
137
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Prove $(-a)^{-1} = -a^{-1}$
This is my proof for part (17) of Lemma 2.3.2 in Bloch's Real Analysis. I'd like if someone verifies that I did not miss or skip a step, did anything unjustified, or anything of this sort. I will ...
1
vote
0
answers
59
views
Real Analysis - Prove that there exists $n, m \in \Bbb{N}$ such that $2m\pi + \frac{\pi}{2} - \epsilon < n < 2m\pi + \frac{\pi}{2}$.
This is a claim that I had made while finding the supremum of $(\sin(n))_{n \in \Bbb{N}}$. The supremum is $1$ if there exists $n \in \Bbb{N}$ such that for all $\epsilon' > 0$, $1-\epsilon' < \...
- 4,335
0
votes
1
answer
35
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Example of basis
Excuse me , can you see this question
, the collection of all open intervals (a,b) together with the one-point sets {n} for all positive and negative integers n is a base for a topology on a real ...
- 93
2
votes
1
answer
87
views
Alternative proof of $a\times0= 0$
I was trying to find a proof of $a\times0 = 0$ by myself (assuming commutativity, associativity, distributivity, etc) and I came up with $$ a+0=a(1) \implies 1 = \frac{a+0}{a} = \frac aa + \frac 0a = ...
2
votes
1
answer
322
views
Bad Proof? Between any two reals is a rational number
I know about the proof found here: Proof there is a rational between any two reals.
I wanted to know if this similar proof is also correct?
Assume $x > 0$. Since $y > x$, it follows $y-x>0$....
1
vote
0
answers
67
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Proof: For any subsequence $a_{n_k}$ Prove $\liminf_{n\to\infty} a_n \le \lim_{k\to\infty} a_{n_k} \le \limsup_{n\to\infty} a_n$
For any convergent subsequence $a_{n_k}$ of $a_n$, Prove: $$\liminf_{n\to\infty} a_n \le \lim_{k\to\infty} a_{n_k} \le \limsup_{n\to\infty} a_n.$$
My attempt
For this proof it should be noted that $...
- 70
3
votes
1
answer
128
views
Exercise about sub-$\sigma$-algebra of $\mathcal{B}(\mathbb{R})$
Let $C=\{(-a, a): a \in \mathbb{R}\}$ and $F=\sigma(C)$.
Prove that $F=\mathcal{B}(\mathbb{R})\cap\{A\subseteq\mathbb{R}: A=-A\}$.
I don't have problems in proving $F\subseteq \mathcal{B}(\mathbb{R})\...
- 319
1
vote
2
answers
349
views
Explain this "contradiction" of the proof of $x > 0$ iff $x \in \mathbb{R}^{+}$
I am working through Apostol's Calculus I and just read about the order axioms. He presents a new undefined concept called positiveness, gives the axioms and then defines symbols $<, >, \leq, \...
4
votes
2
answers
12k
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Proof clarification - If $ab = 0$ then $a = 0$ or $b =0$
I came across a proof for the following theorem in Apostol Calculus 1. My question is regarding (1) in the proof, why is this part necessary? I don't see why you can't begin with (2)
Theorem 1.11
If ...
0
votes
2
answers
92
views
Validity of Proof for 'Possibility of Subtraction' from Apostol 1
I attempted a proof before reading the solution Apostol provides. I don't think it is valid but I am trying to determine why
Theorem 2, Possibility of Subtraction: Given $a \text{ and } b$, there is ...
3
votes
2
answers
69
views
Regarding $x < y \Rightarrow x^n < y^n$ proof rigor.
I came across the implication
$$x < y \Rightarrow x^n < y^n$$
$$x,y>0, n\in Z^+$$
in a textbook and came up with the following proof.
Proof
Since $x<y$ the following chain of inequalities ...
- 33
0
votes
0
answers
56
views
Proving closed interval [0,1] has same cardinality as Real Numbers [duplicate]
I want to prove that the closed interval [0, 1] has same cardinality as Real Numbers. I was able to figure out (0, 1), but need help with proving the closed part. What I have so far is:
the function $...
- 819
2
votes
2
answers
3k
views
If sup A $\lt$ sup B show that an element of $B$ is an upper bound of $A$
(a) If sup A < sup B, show that there exists an element of $b \in B$ that is an upper bound for $A$.
I have argued that if sup A $\lt$ sup B, then choose an $\epsilon>0$ such that sup A +$\...
- 3,306
1
vote
0
answers
1k
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Let $a < b $ be real numbers and consider set T = $\mathbb Q \ \cap \ [a,b].$ Show $\sup \ T =b$
Let $a < b $ be real numbers and consider set T = $\mathbb Q\cap [a,b].$ Show $\sup T =b$
I needed help checking if my proof is correct. If it isn't correct can you please provide the correct ...
- 3,306
3
votes
2
answers
807
views
Proving that a sequence converges to L
Given a sequence $(a_{n})_{n=1}^{\infty}$ that is bounded. Let $L \in R$. Suppose that for every subsequence $(a_{n{_{k}}})_{k=1}^{\infty}$ , either $$\lim_{k \to \infty}a_{n{_{k}}} = L$$
or $(a_{n{_{...
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17
votes
2
answers
19k
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Proof there is a rational between any two reals
This is a problem from Rudin, but I wanted to add my own intuition to it. It uses Rudin's definition of Archimedean property. I'd just like to know if my version holds
If $x \in \mathbb R$, $y\in \...
- 1,755
0
votes
1
answer
138
views
Proving $\mathbb{R}^\mathbb{N}$ ~ $\mathbb{R}$
I am trying to show $\mathbb{R}^\mathbb{N}$ ~ $\mathbb{R}$ using the Cantor Bernstein method. Here is my proof so far:
Let $f: \mathbb{R}\to\mathbb{R}^\mathbb{N}$ be defined as for each $r_n\in\...
- 1,088
-1
votes
2
answers
145
views
prove quadratic polynomial has no real roots
The problem asks me to prove that a polynomial $f(x)=x^2+ax+b$ has no real roots for some $a,b \in \Bbb{R}$
I started by assuming that $f(x)=x^2+ax+b$ has real roots and therefore the determinant $a^...
- 689
0
votes
2
answers
308
views
Conceptual question about trichotomy of real numbers
I was thinking that it is possible to answer this Math.SE question here using the trichotomy of the real numbers but I got some logic trouble.
The question was. Prove that for all $x,y > 0$ ...
- 2,512
1
vote
0
answers
80
views
How to prove that, for any sequence $(s_n)$ of real number and any real number $z$, the following $2$ statements are equivalent?
How to prove that, for any sequence $(s_n)$ of real number and any real number $z$, the following $2$ statements are equivalent?
$1.$ Every subsequence of $(s_n)$ has a further subsequence that ...
- 2,756
6
votes
4
answers
11k
views
How to prove that every real number is the limit of a convergent sequence of rational numbers?
Here is my procedure:
so we want to prove $\forall r\in \mathbb{R},$ there exists a sequence $q_n$ of rationals such that $\forall\epsilon\gt 0,$ there exists a $N$ such that $n\gt N\implies |q_n-r|\...
- 2,756
0
votes
1
answer
67
views
Trouble proving the floor function is onto with the domain being all real numbers
I need to prove that for the mapping $f : \mathbb{R} \mapsto \mathbb{Z} $ given by $ f(x) = \lfloor x \rfloor$, $f$ is onto. I know how I would do it if both the domain and codomain were both $\mathbb{...
3
votes
2
answers
3k
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How to prove the power set of the rationals is uncountable?
Recently a professor of mine remarked that the rational numbers make an "incomplete" field, because not every subsequence of rational numbers tends to another rational number - the easiest example ...
- 7,252
0
votes
0
answers
20
views
Help proving $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $
I am trying to formally prove:
$ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $
where n is an integer, and x and y are natural numbers.
It is obvious that, when $\frac xy$ is ...
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7
votes
5
answers
2k
views
Is this direct proof of an inequality wrong?
My professor graded my proof as a zero, and I'm having a hard time seeing why it would be graded as such. Either he made a mistake while grading or I'm lacking in my understanding. Hopefully someone ...
- 81
0
votes
2
answers
99
views
Cardinality proof verification
Problem:
Let $C \subset (0,1)$ be the set of all numbers whose unique decimal representation contains the number seven. Show that the number of elements in $C$ must be the same as the number of ...
- 422
1
vote
1
answer
54
views
My proof of: $|x - y| < \varepsilon \Leftrightarrow y - \varepsilon < x < y + \varepsilon$
Is it reasonable to prove the following (trivial) theorem?
If yes, is there a better way to do it?
Let $x, y \in \mathbb{R}$.
Let $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$.
$\textbf{...
- 399
0
votes
1
answer
104
views
$xy \le xz$ if both $y \le z$ and $0 \le x$. (very easy proof exercise)
As an exercise, I tried to prove the following theorem.
Please share your thoughts about what I wrote.
(The proof only uses the utensils which are listed below.)
Theorem
Let $x,y,z \in \mathbb{R}$.
...
- 1,321
3
votes
1
answer
2k
views
$x^n y^n = (xy)^n$, proof exercise
As an exercise, I tried to prove the following theorem.
Please share your thoughts about what I wrote.
(The proof only uses the utensils which are listed below.)
Theorem
\begin{equation*}
x^n y^n ...
- 1,321
1
vote
2
answers
4k
views
An upper bound $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$
Problem:
Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$.
Prove that $u$ is the supremum of $A$
if and only if for all $\epsilon > 0$ there is an $a \in A$ such that
$u-\epsilon &...
- 6,684
2
votes
3
answers
75
views
If $x>y$, then $\lfloor x\rfloor\ge \lfloor y\rfloor$, formal proof
For x ∈ ℝ, define by: ⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋).
Claim 1.1: ∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋.
Assume, x, y ∈ ℝ # Domain assumption
...
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2
votes
1
answer
398
views
My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.
What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style.
Theorem
Among three elements of ...
- 1,321
3
votes
0
answers
190
views
My first simple direct proof (very simple theorem on real numbers). Please mark/grade.
What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style.
Theorem
Let $I = [a,b]$ be a non-empty closed ...
- 1,321
1
vote
1
answer
125
views
Possible book correction or am I missing something?
Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
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