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.

learn more… | top users | synonyms

0
votes
1answer
39 views

Proof for the coefficient of $x^n$ in $(x^0 + x^1 + \dots + x^n)^n$

Theorem: The coefficient of $x^n$ in $(x^0 + x^1 + \dots + x^n)^n$ is $\binom{2n-1}{n-1}$. How to prove this? Multinomial theorem produces the following $$ \left(\sum_{k=0}^{n} x^k \right)^n = ...
23
votes
2answers
853 views

A false conjecture by Goldbach

In 1752 Goldbach send this conjecture to Euler: "Every odd integer can be written in the form $p+2a^2$ where $p$ is a prime or $1$ and $a$ is a natural number (can be even 0)." This conjecture turned ...
2
votes
1answer
66 views

Can I assume that the dimension of a vector space is always non-negative?

I'm trying to prove that if $V$ is finite-dimensional and $U_1,...,U_m$ are subspaces of $V$, then $\dim(U_1+...+U_m)\le \dim U_1+...+\dim U_m$ through induction. For $m=1$, the inequality is trivial ...
1
vote
1answer
22 views

Zero of holomorphic function

Let $\Omega \subset \mathbb{C}$ be an open set that contains the unit ball $D$ and let $f \in \mathcal{O}(\Omega)$ a non constant map s.t. $|f(z)| = 1$ for all $z \in \partial D$. Show that $f$ has a ...
3
votes
1answer
74 views

Irrationality of ${5^{1/7}}$

I am struggling with elementary proofs, and would appreciate any feedback as to the logic and structure of my work. Show that ${5^{1/7}}$ does not represent a rational number. Suppose ${5^{1/7}}$ is ...
1
vote
2answers
78 views

Prove the Identity $\pi=2- \sum_{1}^{∞} \frac{(-1)^m}{m^2-\frac{1}{4}} $

By considering the fact that $f(\pi/2)=1$, prove the identity $\pi=2- \sum_{1}^{∞} \frac{(-1)^m}{m^2-\frac{1}{4}} $ This question was is a subsection in a chapter on Fourier series, can I use my ...
1
vote
0answers
36 views

Uniform Continuity of $\frac {1}{x}$ on [$a, \infty$) for positive $a$

$\frac {1}{x}$ behaves nicely in that it's monotone and the derivative is monotone also. So on [$a,\infty$] it can be seen that the $\delta$ which will work everywhere is the $\delta_1$ at the end ...
1
vote
1answer
38 views

Proof strategy/writing for change of variables

Claim: If $f(x)=g(x)$ for all $x$, then $f(x+c)=g(x+c)$ for all x. Proof (attempt): Set $u=x-c$, and substitute $x=u+c$. $f(x)=g(x)$ implies $f(u+c)=g(u+c)$ for all $u$. Because $u$ is a dummy ...
2
votes
1answer
45 views

Prove that it is NOT true that for every integer $n$, 60 divides $n$ if and only if 6 divides $n$ and 10 divides $n$.

This is Velleman's exercise 3.4.26 (b): Prove that it is NOT true that for every integer $n$, 60 divides $n$ iff 6 divides $n$ and 10 divides $n$. I do understand that a number will be ...
4
votes
2answers
59 views

Finding all solutions of $x^2+2x-15\equiv0 \pmod{105}$- Proof strategy.

Find all solutions of $x^2+2x-15\equiv0 \pmod{105}$. Now, I wanted to suggest a proof relying on the algorithm presented in class, and there are some parts where I could use some help or criticism. ...
-2
votes
1answer
51 views

Proof: A matrix with $m$ rows and $n$ colums has $nm$ entries.

How to prove rigorously the following statement: A matrix (a collection of numbers $a_{ij}:1\leq i \leq m, 1\leq j \leq n)$ with $m$ rows and $n$ colums has $nm$ entries. By rigorously I mean ...
1
vote
1answer
40 views

Requirement of a formal proof

There exists a continuous function $f$ whose domain is $[2,5]$ and the range is $(3,4)$. We have to prove that there exists at least one point $p \in (2,5)$ such that $f(p)=p$. Now this is easy to ...
2
votes
6answers
128 views

How to prove $\lim_{n\rightarrow\infty}nx^n=0$ without L'Hôpital's rule, where $x \in [0,1)$??

How to prove $$\lim_{n\rightarrow\infty}nx^n=0$$ without L'Hôpital's rule? where $x \in [0,1)$ and $n=1,2,3,...$. I know one of way to prove this is to treat $n$ is real, and $n$ and ...
0
votes
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
126 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
59 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
33 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
109 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
39 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
177 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
33 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
37 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). ...