For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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0
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2answers
84 views

Prove that $\lim _{x \to \infty} \sin x$ doesn't exist (using delta epsilon)

though there is a question already asked in this site similar to this i want to prove that $\lim _{x \to \infty} \sin x$ doesn't exist using epsilon and delta. I don't know how to do this because ...
3
votes
2answers
123 views

Slick proof that if an open set contains $\mathbb Q$ it has all irrational numbers, except a countable amount.

Basically I need help in proving that if $U\supseteq \mathbb Q $ is an open set in $\mathbb R$ with the usual topology then $\mathbb R \setminus U$ is countable. I'm not really sure how to proceed. ...
1
vote
1answer
23 views

Determining Injectivity, surjectivity, bijectivity, and inverses

I was given a question that begins like this. Suppose that $A$ is the set $\{a,b,c\}$ (these are just names for some three elements - you don't know anything about $a,b,$ or $c$). Consider the ...
1
vote
2answers
55 views

Prove $|a+b+c| \leq |a| + |b| + |c|$ for all $a,b,c \in \mathbb{R}$.

Here is the proof that I am currently working on. Prove $|a+b+c| \leq |a| + |b| + |c|$ for all $a,b,c \in \mathbb{R}$. Hint: Apply the triangle inequality twice. Do not consider eight cases. I ...
1
vote
4answers
98 views

IF $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$

Given $a_n$ be a sequence and IF $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$ I do not know how to do this. Can someone help me with this? Thanks ...
4
votes
0answers
60 views

How to prove the Riemann hypothesis holds for the first non-trivial zero? [duplicate]

The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function $\zeta(z)$ lie on the critical line $\Re(z)=1/2$. The MathWorld page on this topic mentions that the hypothesis ...
2
votes
0answers
18 views

The domain of the sum rule(probability: The logic of science)

Anyone read Probability Theory: The logic of Science. Please help, I've been really stuck for ages at how the sum rule has it domain derived and I don't have any teacher to ask. Question How is ...
1
vote
2answers
37 views

Is this function a bijection?

$f: \Bbb N \to P(\Bbb N)$ be given by $f(n) = \{n+1,n+2,n+3,\ldots\}.$ From general intuition and reasoning I think the function is not injective here is my work. If $n = 1$ $f(1) = ...
0
votes
2answers
38 views

Proof: $ A - (B - C) \subseteq (A - B) - C$

Question: Prove or disprove the following statements: For all sets $A, B, C$: a) $A - (B - C) \subseteq (A - B) - C$ b) $(A - B) - C \subseteq A - (B - C)$ c) If $A - (B - C) \subseteq ...
0
votes
2answers
43 views

Let $(s_n)$ be a convergent sequence of real numbers such that $s_n \neq 0$ for all $n \in \mathbb{N}$ and $\lim_{n \to \infty}s_n=s\neq 0$.

Prove that $\sup \{\frac{1}{|s_n|} : n \in \mathbb{N}\}>0$ Any help on getting this proof started would be appreciated. I know it must be related to proving that $\inf \{|s_n|:n \in ...
1
vote
5answers
49 views

Help with proof that $\sum_{n \in \Bbb{N}} \frac{1}{an + b}$ also diverges?

We know that $\sum_{n \in \Bbb{N}} \frac{1}{n}$ diverges. So it seems likely that $\sum_{n \in \Bbb{N}} \frac{1}{a n + b}$ will for any real $a, b$. I'm having trouble proving it just for the ...
1
vote
0answers
32 views

The Polar-Coordinate Form of Cauchy-Riemann

Write $f(z) = u(r, \theta) + iv(r, \theta)$; suppose that the first-order partials of $u, v$ with respect to $r, \theta$ are continuously differentiable in some neighborhood of $z$ and satisfy ...
1
vote
1answer
57 views

$\forall x \in \Bbb Q, \exists y \in \Bbb Q$ so that $x + y \in \Bbb Z $

Let $\Bbb Q$ be set of all rational numbers. Proof: $\forall x \in \Bbb Q, \exists y \in \Bbb Q$ so that $x + y \in \Bbb Z $ This statement is true. Here is a proof: Suppose $x$ is some rational ...
1
vote
5answers
56 views

For all sets $A$, $B$, and $C$, if $A-B \subseteq A - C$ then $ A \cap C = \varnothing $

Prove the statement P: For all sets $A$, $B$, and $C$, if $A-B \subseteq A - C$ then $ A \cap C = \varnothing $ My attempt to answer: This statement is true, and here is a proof: Proof: ...
1
vote
3answers
49 views

How can I prove that if $\lim_{n \to \infty}s_n=s$ then $|s_n-s|< \epsilon$ is equivalent to $s-\epsilon <s_n <s+ \epsilon$

My professor casually mentioned this in class and told us to prove it if we weren't convinced, however, I cannot find how to prove it.
0
votes
4answers
40 views

Proving binomial coefficient formula based on Pascal's triangle

I am trying to practice proving things, and I came across one I wasn't sure about. We already know that $\binom{n}{k}$ is the sum of the two corresponding "parent" entities in Pascal's triangle, ...
1
vote
3answers
46 views

Let $a,b \in \mathbb{R}$. Show if $a \leq b+\frac{1}{n}$ for all $n \in \mathbb{N}$, then $a \leq b$.

I was able to prove this using sequences, however, I was told that there is another prove that does not use sequences and I cannot figure that one out. How can I prove this without using sequences? ...
1
vote
2answers
28 views

Listing all elements of a set [duplicate]

I was given a question like the following: Let $A = \Bbb Z$, $B = [-1,\pi]$ , $C=(2,7)$. List all Elements of $A \cap (B^c \cap C)$. I do not really understand how to got about this problem. I ...
0
votes
1answer
33 views

Straight line through $(a,b)$ with slope $m$ is the graph of the function $f(x) = m(x-a) + b$

Spivak's Calculus Chapter 3 Problem 6 says: Show that the straight line through $(a,b)$ with slope $m$ is the graph of the function $f(x) = m(x-a) + b$. Since the slope in a graph of a line is ...
0
votes
1answer
21 views

Determining the image of a function [duplicate]

I was given a function that says: What is the image of the function $F: \Bbb Z \times \Bbb N \rightarrow \Bbb R$ given by $f(a,b) = \frac{(a-4)}{7b}$ I need help really understanding how to find an ...
0
votes
1answer
26 views

Show that $\left \{ \bigcup_{i\in I}A_{i}:I\subseteq \{1,\dots, n\} \right \}$ is a $\sigma$-algebra

Let $\{A_{i}\}_{i = 1}^{n}$ be a family of pairwise disjoint subsets of $X$. It is said that $$\mathcal{F}:=\left \{ \bigcup_{i\in I}A_{i}:I\subseteq \{1,\dots, n\} \right \}$$ is a $\sigma$-algebra. ...
4
votes
1answer
50 views

Do proof assistants like Coq really need to actually perform computations to prove n <= m, or is there a more optimal algorithm?

For example, trying to prove that 100,000 <= 1,000,000. But Coq has a stack overflow, meaning it's actually trying to perform the 100k computations. ...
-8
votes
2answers
108 views

Is this map proof that Four Color Theorem is wrong, or I'm missing something? [closed]

Yesterday, after hours of trying I developed one map for which I could not find solution with 4 colors, so I opened topic to ask is there solution for map, and it turned out, there was been solution(I ...
0
votes
1answer
55 views

Proving $\pi$ irrational: help with Lambert's proof. “Circularity”?

This expression is irrational. $$\tan(x)=\frac{x}{1-\frac{x^2}{3-\frac{x^2}{5-...}}}$$ But then he used the fact that $\tan{\frac{\pi}{4}}=1$, so $\frac{\pi}4$ is irrational. But how can we use ...
1
vote
2answers
24 views

Is it possible to prove that the gradients of the real and imaginary parts of a complex analytic functions have the same length?

Suppose I have a complex analytic function $$f(x,y)=(x+iy)^n$$ where both $x$ and $y$ are real and $n$ is an integer. Is it possible to prove that the gradient of the real part of $f$ and the ...
2
votes
1answer
48 views

How To Prove Irrational Square Roots and Inequalities In Courant's Calculus Book? [closed]

Here's the proofs questions in a screenshot The first questions ask about proving the irrationality of non perfect squares. Numbers 3,5, and 6 ask for inequality proofs. I find it daunting that the ...
0
votes
2answers
38 views

Verifying a Proof for Spivak's Calculus Question (Chapter 2 Problem 9)

It says "Prove that if a set $A$ of natural numbers contains $n_0$ and contains $k+1$ whenever it contains $k$, then A contains all natural numbers $\ge n_0$". Am I allowed to construct another set ...
0
votes
1answer
34 views

solving for the inductive step in a proof by induction

I have no trouble solving for the base case. I need help solving the inductive step. I know that the nth line creates n new regions. But I don't know if that's based on intuition or if I have to ...
-1
votes
3answers
177 views

Four color theorem, What did I miss?

I am not saying that I have proven Four color theorem to be wrong, either I am saying that four-color theorem is wrong but I got one idea so I want to know what I am missing ( This is not proffesional ...
1
vote
1answer
20 views

Proof: superharmonic function equal on $\partial D$ and at one point inside of D to its harmonic function, is harmonic on D (D compact)

I am looking for a proof (literature or short idea) for the following statement, which I have found in several sources: Let $M$ be a riemannian manifold, let $f:M\to\mathbb{R}$ be a superharmonic ...
10
votes
1answer
175 views

Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for-

Let $n$ be a positve integer. Prove that$$\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$$ for each non-negative ...
3
votes
3answers
36 views

reflexive, symmetric, and transitive relations proof

Let $A = \{1, 2, 3, ... , n\}$ where $n$ is a positive integer. Let $F$ be the set of all functions from $A$ to $A$. Let $R$ be the relation on $F$ defined by: for all $g, f \in F, fRg$ if and only ...
-4
votes
1answer
32 views

Proof reflexive, symmetric and transitive relations

Let $A = \{1, 2, 3, ... , n\}$ where $n$ is a positive integer. Let $F$ be the set of all functions from $A$ to $A$. Let $R$ be the relation on $F$ defined by: for all $g, f \in F, fRg$ if ...
1
vote
0answers
69 views

Proving that $b^6-2a^3b^3-2b^3+a^6-2a^3+1$ is never a nontrivial integer square?

I'm trying to prove that if $a,b$ are integers, and $b^6-2a^3b^3-2b^3+a^6-2a^3+1$ is a square [integer], then $ab=0$. What general tools are available to attack such a problem?
1
vote
2answers
35 views

Proof of onto and one-to-one functions, composition

I want to prove this: Let $f: A \to B$ and $g: B \to C $ be functions. if $g \circ f$ is onto, and $g$ is one-to-one, then f is onto. Here is what I have done, can someone please verify my work: ...
1
vote
1answer
25 views

Computing the GCD

So I was given multiple questions of computing the GCD of $\gcd(10;45)$ and $\gcd(1701;3768)$, etc. The questions generally worked with numbers and I was able to solve it quite simply since I knew ...
4
votes
1answer
31 views

Help proving generalized Jensen's inequality $\mathbf{E}[f(\cdot,X(\cdot))\mid \mathscr{G}] \geq f(\cdot,\mathbf{E}[X\mid\mathscr{G}](\cdot))$

I'm reading Meyer's seminal work Probability and Potentials (1966), in which he states the following "borrowed" theorem from Dubins "Rises and Upcrossings of Nonnegative Martingales" (1961). ...
1
vote
1answer
31 views

Proving that $O(n)$ is compact

Let $O(n)$ denote the group of orthogonal matrices under multiplication. We want to show that this is set is compact. To show $O(n)$ is compact, we can use Heine-Borel and show that it is closed and ...
5
votes
1answer
170 views

Conjectured compositeness tests for $N=k \cdot 2^n \pm c$

How to prove that these conjectures are true ? Definition : $\text{Let}~ P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)~ , \text{where}~ m ...
3
votes
4answers
98 views

Prove that $n!>n^2$ for all integers $n \geq 4$.

I am working on induction problems to prep for Real Analysis for the fall semester. I wanted proof verification and editing suggestions for part (a), and assistance understanding part (b). For part ...
2
votes
1answer
47 views

Let p<q both be prime numbers. Prove that log is not rational number

So i was given a question that starts off like this Prove that $\log_q(p)$ is not a rational number. Recall that $\log_y(x)$ for real numbers $x,y>0$ is defined to be the real number $r$ so ...
0
votes
1answer
27 views

Computing the gcd of a relatively prime polynomial

I was given a question that starts off like this. Suppose that $a, b \in \mathbb{N}$ and relatively prime. For each of the following, if the answer must be one particular number, then compute it; ...
0
votes
1answer
27 views

Proving the set of finite subsets of $\mathbb{N}$ is countably infinite [duplicate]

So I was given a question that begins like this. Let $P_{\text{fin}}(\mathbb{N})$ be the following set (called the finite power set of $\mathbb{N}$): $$ P_{\text{fin}}(\mathbb{N}) = \{X ...
0
votes
1answer
17 views

Determining cardinality and inverse

Let the function $\chi: P(Z) \to P(Z)$ be defined by $\chi(B) = B^c$ for any $B \in P(Z)$. (In other words, $\chi$ sends a subset $ B \subseteq Z$ to its complement, $B^c$, i.e. the set $Z - B$.) ...
0
votes
2answers
39 views

Uniform Convergence and limit $(n+1)\int_0^1 x^nf(x) \; dx$ [duplicate]

If $f$ is a continuous real-valued function, show that $$ f(1)=\lim_{n\to \infty} \int_0^1 (n+1)\,x^n \,f(x) \; dx $$ I am looking for a general hint or steps to proceed but I want to fill them in. ...
0
votes
2answers
28 views

Proving a number can be chosen to a multiply a $\mathbb{R}^2 \rightarrow \mathbb{R}$ function so that it never exceeds another function

Let $p,q,r$ be real numbers with $p,r,pr-q^2 > 0$. I am trying to prove $\exists\gamma> 0$ s.t. $\forall(x,y) \in \mathbb{R}^2 . px^2 + 2qxy + ry^2 \geq \gamma(x^2+y^2).$
9
votes
6answers
164 views

Follow-up Question: Proof of Irrationality of $\sqrt{3}$

As a follow-up to this question, I noticed that the proof used the fact that $p$ and $q$ were "even". Clearly, when replacing factors of $2$ with factors of $3$ everything does not simply come down to ...
2
votes
2answers
58 views

Prove that $\exists z \in \mathbb R\forall x \in \mathbb R^+[\exists y \in \mathbb R(y-x=y/x) \iff x \neq z]$

This is Velleman's exercise 3.4.13: Prove that $\exists z \in \mathbb R\forall x \in \mathbb R^+[\exists y \in \mathbb R(y-x=y/x) \iff x \neq z]$. I am am stuck on that one. Seems like I am ...
1
vote
2answers
78 views

Explain the proof of irrationality of $\sqrt{2}$

How does this proof show the irrationality of $\sqrt{2}$ ? I am new to proofs and don't really understand the logic used here.
3
votes
7answers
117 views

Proving $\frac{1}{1\cdot3} + \frac{1}{2\cdot4} + \cdots + \frac{1}{n\cdot(n+2)} = \frac{3}{4} - \frac{(2n+3)}{2(n+1)(n+2)}$ by induction for $n\geq 1$

I'm having an issue solving this problem using induction. If possible, could someone add in a very brief explanation of how they did it so it's easier for me to understand? $$\frac{1}{1\cdot3} + ...