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
54 views

Prove the convergence of a sequence involving integrals

I need to prove the following: Assume $f:[a,b] \to R$ is continous, $f(x)\leq0$ for all $x \in [a,b]$, and $M=sup\{f(x):x \in [a,b]\}$.Show that: $$\{[\int [f(x)]^{n}dx]^{1/n}\} \to M$$ This result ...
0
votes
1answer
31 views

“Natural” Homeomorphisms, Retracts and Knots

I have been trying to prove the following result for a few days now, and have made some amount of progress, but now I'm struggling. This is what I'd like to prove: Every proper knot, K has a retract ...
0
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0answers
43 views

Why is this kernel isomorphic to $\Omega^{n-2}(E-2D)$?

I'm reading this article and I didn't understand the proof of the item (1) of this proposition on page 225 (see below): I have the following questions: I didn't understand how these inclusions ...
7
votes
2answers
109 views

An alternative proof for sum of alternating series evaluates to $\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$

How does one prove the given series? $$\sum_{n=0}^\infty\left(\frac{(-1)^n}{4n-a+2}+\frac{(-1)^n}{4n+a+2}\right)=\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$$ This series came up in xpaul's ...
1
vote
1answer
49 views

Suppose that $(s_n)$ converges to s. Prove that $(s_n^2)$ converges to $s^2$

I am really struggling with my proofs class, I don't really understand how to prove a statement like this, or what the epsilon is standing for.. Suppose that $(s_n)$ converges to s. Prove that ...
1
vote
1answer
76 views

Let G be a group of order 2n with n being an odd integer. Prove that G has a normal subgroup of order n.

I have been trying to work this problem out for a bit and am stuck. Does anybody have any ideas how to proceed or solve this? I think this has something to do with rings and fields but I can't seem ...
2
votes
6answers
269 views

Disproving existence of real root in some interval for a quintic equation

Disprove the statement: There is a real root of equation $\frac{1}{5}x^5+\frac{2}{3}x^3+2x=0$ on the interval (1,2). I am not sure whether to prove by counter-example or by assuming the statement is ...
1
vote
1answer
105 views

Proof of easy matching condition for Hall's theorem

I was studying with the recitations provided in the course 6.042 "Mathematics for Computer Science" of MIT OCW and while studying the proof of Hall's marriage problem, I understood the first proof ...
-3
votes
1answer
46 views

Proof By Induction - Factorials

$\left(\forall n \in \mathbb{N}\right)\left((n + 1)! = (n + 1) \cdot n!\right)$ Prove the following statement by induction: for all $n \in \mathbb{N}$ $\sum_{k=0}^{n}(k \cdot k!) = (n + 1)! − 1$ ...
0
votes
2answers
83 views

Arbitrary Set A a Function??

Assume you have an arbitrary set A, let RA be the relation defined on A × Power Set(A) by, for all a ∈ A and B ⊆ A, "a RA B iff a ∈ B" 1.Let A = {0, 1}. Is RA a function? Justify? 2.Find a set A ...
0
votes
3answers
54 views

Functions Proof

Consider the function f : N → N defined, for every n ∈ N, by "f(n) = (n + 1)! − 1" 1.Prove that, for every n ∈ N, f(n + 1) > f(n) 2.Prove that f is injective. Can somebody shed some light on ...
2
votes
2answers
65 views

Prove complex equation

Prove the following $$\frac{1}{z-1}*\frac{1}{z^n}= \dfrac{1}{z-1} - \sum_{k=1}^n\frac{1}{z^k}$$ for any integer n greater than 0. DO NOT USE .... I believe that I can use mathematical induction. ...
2
votes
1answer
102 views

Prove the summation involving Stirling numbers of the first kind

I have been trying to prove or disprove this for 2 days now, but i don't even know where to begin. $$ 1 = \sum_{m=1}^{n} \sum_{k=1}^{n} \frac{x^{n-m}(-1)^{n-k-m} \left[\matrix ...
5
votes
4answers
399 views

Give a combinatorial argument

Give a combinatorial argument to show that $$\binom{6}{1} + 2 \binom{6}{2} + 3\binom{6}{3} + 4 \binom{6}{4} + 5 \binom{6}{5} + 6 \binom{6}{6} = 6\cdot2^5$$ Not quite where to starting proving this ...
5
votes
1answer
75 views

What would happen to Bernoulli's inequality if $x<-1$?

Bernoulli's inequality says that $(1+x)^n \geq 1+nx$ for all $x > -1$ and for all $n \in \mathbb{N}$. The questions asks $what \ can \ you \ say \ if \ x\le-1 \ ?$ So I was just trying out ...
0
votes
3answers
43 views

Show $F_n$ has a least element and it is prime. [closed]

let $F_n$ be the set of positive factors of n greater than 1 and $n\in \mathbb{N}$. Show $F_n$ has a least element and it is prime.
1
vote
1answer
92 views

Number of Necklaces of Beads in Two Colors

I was reading this paper, and came across an equation which gives an expression for the number of necklaces of beads in two colors, with length n. $Z_n = \dfrac{1}{n} \displaystyle \sum \limits_{d ...
1
vote
1answer
40 views

Proof by induction that x_n < 2^n

Define $x_1=x_2=x_3=1$ and $$x_{n+1}=x_n+x_{n-1}+x_{n-2}$$ for $n≥3$. Prove that $$x_n<2^n \qquad ∀n\in\mathbb N$$ I have shown the base case and supposed the result in the inductive case. Then, ...
1
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0answers
52 views

Proving multiplicative property of euler's totient function $\phi$ using probability

If $m,n$ are co-prime , we know that $\phi(mn)=\phi(m)\phi(n)$. I want to prove it using probability. Probability that a selected number less than or equal to $mn$ is co-prime to $mn$ = ...
1
vote
1answer
58 views

Does the orthocenter really exist for all triangles?

Imagine I draw a triangle on a cartesian plane so that it has vertices at $P=(0,0), Q=(b,c), R=(a,0)$ where $b<a$ and $b,a>0$. As far as basic logic goes we should be able to arrange all ...
1
vote
3answers
51 views

If homogeneous system has infinitely many solutions, does every system with that left hand side?

Hi I am trying to work on a proof question that I took note of because the answer did not come to mind immediately. I also found it was a good question to illustrate something I do not understand very ...
1
vote
2answers
56 views

Demonstrating that an Eulerian path can exist.

For every pair of bus stops in a city there is only one one-way route between them, now demonstrate the existence of a maximal path routing through every stop.
3
votes
2answers
61 views

Proving facts regarging a graph with a degree of $n$ and no cycles of length more than 3.

Let H be a simple graph that has no cycles of length more than 3. Each vertex has degree of n $n$. Is it possible to prove H has at least $2n$ vertices?
2
votes
2answers
70 views

Find the probability that a geometric random variable $X$ is an even number

Let $\alpha$ be the probability that a geometric random variable $X$ with parameter p is an even number a) Find $\alpha$ using the identity $\alpha=\sum_{i=1}^{\infty}P[X=2i]$ b)Find $\alpha$ by ...
1
vote
0answers
41 views

Greatest common divisor / euclidean algorithm linear combination proof [duplicate]

Consider integers $m$ and $n$, not both 0. Show that gcd$(m,n)$ is the smallest positive integer that can be written as $am + bn$ for integers $a$ and $b$. I'm confused on what exactly to do--I'm ...
0
votes
1answer
36 views

Euclidean Algorithm / arithmetic mod question

I'm not sure how to approach this problem: Solve for $x$. $788x \equiv 24 (mod 1647)$. I know that if 24 were replaced with 1, I could just do a backwards euclidean algorithm to find x. Can I still ...
1
vote
1answer
53 views

Comparing Statements and predicates using Truth Tables

Consider the four statements: $∃x$ $∀y$ $p(x, y)$ $∃y$ $∀x$ $p(x, y)$ $∀x$ $∃y$ $p(x, y)$ $∀y$ $∃x$ $p(x, y)$ which we call S1, S2, S3 and S4 respectively. Does there exist a predicate p such ...
3
votes
2answers
48 views

Proof of Power of Twos and Threes

Are $(1,2), (2,3), (3,4)$, and $(8,9)$ the only consecutive integers that are a power of two and a power of three? And if they are, how do I prove this?
2
votes
3answers
65 views

Fun problem. Apparently $\prod_i(1-p_i) \geq 1 - \sum_ip_i$ with $p_i \in [0,1]$ is always true. But how to demonstrate it?

so, I want to demonstrate the validity of the following inequality: $$ \prod_i(1-p_i) \geq 1 - \sum_ip_i $$ with $p_i \in [0,1]$, it is always true, which it seems to be always the case if you test ...
1
vote
1answer
133 views

If $f$ is an entire function such that $|Re(f(z))| \leq |z|^n, \forall z,$ then $f$ is a polynomial of degree at most $n$

Let $f$ be an entire function such that $|Re(f(z))| \leq |z|^n, \forall z$ where $n$ is a positive integer. Could anyone advise me how to show $f$ is polynomial of degree at most $n \ ?$ Hints ...
0
votes
1answer
44 views

Proof for $\forall x A \Leftrightarrow \neg \exists x \neg A$

I try to proof, that $\forall x A \Leftrightarrow \neg \exists x \neg A$ I know how to proof, that $\forall x A \Leftrightarrow \exists xA$, but I don't understand, how to get negation.
1
vote
1answer
33 views

Question About Set Relations?

Prove that $\text{Domain}(S\circ R) \subset \text{Domain}(R)$ where $ R $ is a relation from $ A $ to $ B $ and $ S $ is a relation from $ B$ to $ C $. My solution: I suppose that there is an ...
1
vote
1answer
37 views

Properties of periodic complex functions

Suppose $f$ is an entire function and real on the lines $\{z\in\mathbb{C}: \mathrm{Re}(z)=1\}, \{z\in\mathbb{C}: \mathrm{Re}(z)=2\}.$ Suppose $\text{lim}_{1\leq x \leq 2,y \to \infty}f(x,y)=0,$ where ...
0
votes
0answers
50 views

Equalities involving integral of harmonic function

Let $u$ be harmonic on $D(0,r),$ where $r>1.$ Could anyone advise me on how to establish the following equalities: $\begin{align} \int^{2\pi}_{0}u(e^{it})\text{cos}^2(\frac{t}{2})dt=\pi ...
2
votes
0answers
71 views

How to use a very complicated theorem for proving simpler statements without falling into a loop?

There are some too complicated theorems in mathematics which have very complicated proofs in hundreds of pages. There are few mathematicians who are aware of the entire proof of such theorems in full ...
3
votes
1answer
86 views

Application of Riemann Mapping Theorem to conformal automorphism

Let $\Omega=G -\{a,b\},$ where $G$ is non-empty simply connected bounded domain in $\mathbb{C}$ and $a \neq b.$ Could anyone advise me how to find all conformal automorphism of $\Omega \ ?$ I tried to ...
1
vote
3answers
53 views

Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$

Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$. Ok I'm finally understanding what each of these ...
0
votes
2answers
38 views

Recursive definition proof

I'm having trouble proving the following: $a_0 = a_1 = 1$ and $a_n = a_{n-1} + 2a_{n-2}$ for $n \ge 2$. Prove that all the terms $a_n$ are odd integers. It makes sense since an odd number is of the ...
3
votes
3answers
97 views

Why $P(A) \cup P(B)$ is not equivalent to $P(A \cup B)$

I know that they are equivalent if we have $\cap$ instead of $\cup$. If we start from left, we have: $X \in P(A) \cup P(B)$ $X \subset A \lor X \subset B$ If we start from right, we have: $X \in ...
-1
votes
1answer
26 views

How to prove that if $T(n)=\max_{1\le q \le n-1}(T(q)+T(n-q))+\Theta(n)$ then $T(n)=O(n^2)$

I have this recursion relation: $T(n)=\max_{1\le q \le n-1}(T(q)+T(n-q))+\Theta(n)$. How do I show that: $T(n)=O(n^2)$ Thank you!! (P.S. I hope it's OK to ask it, but I'm looking for the simple ...
0
votes
3answers
69 views

How to prove $ 1^3+2^3+… +n^3 = (1 + 2+ \dots +n)^2 $ by induction? [duplicate]

I need to prove that for each natural n: $$ 1^3+2^3+... +n^3 = (1 + 2+ ...+n)^2 $$ How do I do that? how do I know whether I should choose strong induction or simple induction?
2
votes
1answer
195 views

Clarification: Proof of the quotient rule for sequences

My Problem I am currently looking for a proof for the quotient rule for sequences: $a_n$ and $b_n$ are two sequences with the limes a,b. So: When $ a_n \rightarrow a$ and $ b_n \rightarrow b$ ...
1
vote
2answers
683 views

How to derive the volume and surface area formula for truncated cylinder without calculus?

several hours ago, I saw the problem involving geometry. The problem is tell about truncated cylinder. I want to know how to derive the formula of volume and its surface area without calculus, but ...
0
votes
0answers
60 views

Integrals Over Paths and Surfaces proof

Let S be a sphere of radius r and p be a point inside or outside the sphere (but not on S). Show that $\iint\limits_S \frac{1}{\mid x-p \mid}\ dS=4\pi r $ if r is is inside S and $=4\pi r^2/d$, if p ...
3
votes
4answers
233 views

Reference textbook about proof techniques

I am looking for some good recommended reference textbooks about proof techniques. Someone told me "G. Polya - How to solve it" is kind of standard, but quite old. I am looking for a book that ...
0
votes
1answer
23 views

How to solve $T(n)=\frac2n\sum_{i=1}^{n-1}T(i)+\Theta(n)$

I need to solve this Recursion Relation:$$T(n)=\frac2n\sum_{i=1}^{n-1}T(i)+\Theta(n)$$ Do you have any idea how? (I try to find another way instead induction). (We should get: $T(n)=\Theta(n\log n)$) ...
0
votes
0answers
7 views

Prove that $cl_{eq}(R) = (cl_{sym}(R))^*$

Prove that $cl_{eq}(R) = (cl_{sym}(R))^*$ $cl_{eq}(R) = \bigcap\{S | $ S is an equivalence relation and $R \subseteq S\}$ is the equivalence closure of R. $R^* = \bigcap\{S | $ S is reflexive, ...
-1
votes
1answer
23 views

big $\Theta$ question dealing with $\log_2{n}$ and $\log_{10}{n}$

Show that $\log_{10}{n} = \Theta(log_2{n})$. I know that I have to show that 1) $\log_{10}{n} = O(\log_2{n})$ show: $\log_{10}{n} \le C * \log_2{n}$ and 2) $\log_2{n} = O(\log_{10}{n})$ show: ...
0
votes
1answer
55 views

Prove $\sum_{k=0}^{n-1}\frac{n(-1)^{n-k+1}}{n-k}{n-1 \choose k} = 1$

To finish a proof i have been working on i must prove the following: $$ \sum_{k=0}^{n-1}\frac{n(-1)^{n-k+1}}{n-k}{n-1 \choose k} = 1 $$ I have checked that it does work empirically, but of course ...
0
votes
0answers
23 views

Big theta question

Find a number $a$ with $s(n) = \Theta(a^n)$ for $s(n) = 1^{(n^2+200n+5)}$. I worked out that $a = 1$ and that $1^{(n^2+200n+5)} \le C * 1^n, C = 1, n = 0.$ So long as n $\ge$ 0 the right-hand side ...