If you already have a proof for some result, but want to ask for a different proof (using different methods).

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3answers
67 views

I have a question about groups of finite order.

I want to show that if a group G has finite even order then it must have an odd number of elements that are their own inverse, and if G has odd order then it has no elements of order two. I know this ...
2
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1answer
34 views

Solving a divisibility problem using group theory

Inspired by this I decided to show this: Let $P_n=\displaystyle\prod_{1\leq i<j\leq n} \big(x_i-x_j\big)$ where $x_1\,\dots\, x_n$ are arbitrary integers. Prove $n!\,\big|\, 2P_n$. I am ...
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2answers
43 views

A simpler way to prove this trigonometric identity?

The question asks to prove: $$ \frac{\tan{A}}{1-\cot{A}}+ \frac{\cot{A}}{1-\tan{A}}=\sec{A}\csc{A}+1$$ using only: $$ \sin^2{A}+\cos^2{A}=1\;\; \text{ & }\; \;\tan^2{A}+1=\sec^2{A}\;\; \text{ ...
2
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0answers
46 views

Alternative Proof for “Roots of Mertens Function-Farey Sequence-Cosines Relations”

You can write Merten's function as $$ M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} , $$ where $\mathcal{F}_n$ is the Farey sequence of order $n$. The sum may be split into imaginary and real ...
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5answers
69 views

If one of the hypotheses holds, then one of the conclusions holds. (looking for a proof)

Using a huge truth table, I proved the theorem below. I cannot find a more elegant proof. I tried to rewrite expressions; e.g. using the distributive laws and the laws of absorption - to no avail. Is ...
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1answer
48 views

Computing a field extension by hand

Let $k$ be a field, and $K=k(t)$ be the field of rational functions (where $t$ is indeterminate). Let $F=k(t^2)$. A typical element of $F$ will look like: $$ \frac{\displaystyle \sum_{i=0}^{n} a_{2i} ...
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1answer
34 views

Proof that 6 divides $a \in \mathbb{Z}, a(a^2 - 7)$

I am trying to prove a question from my tutorial sheet, is this an acceptable proof? Six cases exist: $$a,k \in \mathbb{Z}, a(a^2 - 7) = 6k \\\text{Proof:}\\ a = 0 \mod 6 \longrightarrow a^2 = 0 \mod ...
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0answers
32 views

Intuition of Newton's identities, is it worth persuing these thoughts? Should be able to show it from special case.

I've "accidentally" stumbled on a special case of it (i=n) and it is very intuitive. Then there's some sort of "transpose" going on with the terms. So I'm thinking there might be something there. I ...
2
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2answers
58 views

Last decimal digit of any perfect square must be $0,1,4,5,6$ or $9$

Last decimal digit of any perfect square must be $0,1,4,5,6$ or $9$ My Proof: Ten cases exist, yielding the following equalities: $$(1\mod{10})^2 = 1\mod{10}$$ $$(2\mod{10})^2 = 4\mod{10}$$ ...
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2answers
79 views

Combinatorics proof of “sum of (k choose m) with k from m up to n is equal to n+1 choose m+1”

I've already proved this statement algrebraically. I'm asked to prove it with combinatorics. So far I came up with, LHS= # ways to choose m apples from a total of m,m+1,...,n RHS= # ways to choose ...
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2answers
50 views

Proof of the 'second' triangle inequality

I am trying to prove the 'second' triangle inequality: $$||x|-|y|| \leq |x-y|$$ My attempt: $$----------------$$ Proof: $|x-y|^2 = (x-y)^2 = x^2 - 2xy + y^2 \geq |x|^2 - 2|x||y| + |y|^2 = ...
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2answers
44 views

Proof: Each common divisor c of a,b divides GCD(a,b)

there already exists a proof for this theorem: http://www.proofwiki.org/wiki/Common_Divisor_Divides_GCD This one, however, uses Bêzout's Identity. I'm not allowed to use this for the proof. So, I ...
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4answers
214 views

Easy proof for sum of squares $\approx n^3/3$

I'd like to prove to my (undergraduate, not math-major) students that $$ \lim_{n\to\infty} \frac{1}{n^3}\sum_{k=1}^n k^2 =\frac{1}{3}, $$ to later show them that this can be interpreted as taking ...
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0answers
43 views

On Lucas Lehmer primality Test

http://primes.utm.edu/notes/proofs/LucasLehmer.html is proof of the Lucas Lehmer Test I read. The part I do not understand is why did he consider the sequence $S_n=S_{n+1}^2-2$. I mean why would ...
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1answer
76 views

An algebraic proof that a set has $2^n$ subsets. (I'm looking for an inductive argument.)

There will be duplicates of this, so let me explain why I am asking: I have become blind to what it may be, so I want hints. I am blind because I can do it "combinatorially". The question wishes me ...
3
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3answers
114 views

Proof without words for $\sum_{i=0}^\infty(-1)^i\frac{1}{2i+1}$

$$\sum_{i=0}^\infty(-1)^i\frac{1}{2i+1}$$ $$1-\frac13+\frac15-\frac17+\frac19-\cdots=\frac\pi4$$ Does anyone know of a proof without words for this? I am not looking a for a just any proof, since I ...
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4answers
123 views

Solution of $\dfrac{a}{b}=\dfrac{a'}{b'}$ if $a,b,a',b' \in \mathbb{N}$

Let $\dfrac{a}{b}=\dfrac{a'}{b'}$ , $a,b,a',b' \in \mathbb{N}$ s.t. $a$ and $b$ have no common factors and it is presumed that $a'>a$ and $b'>b$. How can we show that the only solution to this ...
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2answers
191 views

closed unit ball in a Banach space is closed in the weak topology

Let $V$ be a Banach space. Show that the closed unit ball in $V$ is also closed in the weak topology. I know this is a consequence of the statement any closed convex subset in $V$ is closed in the ...
3
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0answers
52 views

How to give an epsilon-delta proof of this limit statement? [duplicate]

Although I know a couple of proofs of the statement $$ \lim_{x \to 0 } \frac{\sin x}{x} = 1, $$ I would like to be able to come up with a proof using the definition of the limit (i.e. an ...
3
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2answers
71 views

Observations needed to justify an algebraic passage in proof of a property of $\varphi$ (Totient function)

Let $\varphi$ be the Euler's totient function and let $n\in \mathbb{N}$ be factorized in primes as $n=p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_l^{\alpha_l}$. I was looking for alternative methods to ...
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1answer
53 views

Abel's test: how to prove it

Consider the following statement: Let $b_n$ satisfy $b_1 \ge b_2 \ge \dots \ge 0$ and let $\sum_{n=1}^\infty a_n$ be a series for which the partial sums are bounded i.e. there exists $A > 0$ such ...
2
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0answers
87 views

Choices of epsilons in proof : $(b_n) \to b$ implies $\left\{\frac{1}{b_n}\right\} \to \frac{1}{b}$ (Abbott pp 47 T2.3.3.iv) [closed]

Original became long, ergo I ask anew. The trick is to look far enough out into the sequence $(b_n)$ so that the terms are closer to b than they are to 0. Consider the particular value $e_0 = |b|/2$. ...
1
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1answer
61 views

Prove that a function defined on points in a plane is zero

Let $n\ge3$ be an integer, and $f:P\to\mathbb R$ be a function defined on any point in the plane $P$, with the property that for any regular n-gon $<A_1A_2A_3\cdots A_n>$, ...
2
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0answers
118 views

Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we ...
1
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1answer
71 views

Proof without mean value theorem

Is it possible to prove the following without using the mean value theorem: If $f$ is differentiable on an interval containing $0$ and if $\lim_{x \to 0} f'(x) = L$ then $f'(0) = L$. I have ...
13
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1answer
273 views

A beautiful inequality for convex functions

Let $f\in \mathcal{C}([0,1],\mathbb R_+)$ increasing. Prove that there exist $g,h\in \mathcal{C}([0,1],\mathbb R)$, convexs, such that $g\leqslant f \leqslant h$ and : $$\displaystyle ...
17
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8answers
448 views

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
0
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2answers
58 views

Consecutive positive integers proof problem

Consider any three consecutive positive integers. Prove that the cube of the largest cannot be the sum of the cubes of the other two. Work: I tried to prove via contradiction. I made three ...
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0answers
28 views

Dimension Theorem modification

The Dimension Theorem says $$ \dim(U+W) = \dim(U) + \dim(W) - \dim(U \cap W) $$ The proof of this theorem uses the bases of $U$, $W$, and $U\cap W$. Is it possible to prove this theorem with just ...
2
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2answers
99 views

How to prove from the definition that $X_n \xrightarrow{\mathbb{P}} X$ implies $\frac{1}{X_n} \xrightarrow{\mathbb{P}} \frac{1}{X}$?

Let $X, X_1, X_2, \ldots : \Omega \to (0,\infty)$ be random variables such that $X_n \xrightarrow{\mathbb{P}} X$. I'd like to show from the definition that $\frac{1}{X_n} \xrightarrow{\mathbb{P}} ...
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4answers
333 views

How to prove $f\equiv 0$ without Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know we can use Weierstrass theorem but I'd like to ...
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0answers
57 views

Extreme Value Theorem: from 1 variable to several variables?

When studying functions of several variables, one useful idea is to use the same result for one variable. For example, to prove that a local extremum of a function $f(x_1,\dots,x_n)$ is a critical ...
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0answers
94 views

Back-and-Forth Argument vs. “One-Way” Argument

The wikipedia article on the Back and Forth Argument claims at the end: If we iterated only step $(1)$, rather than going back and forth, then in some cases the resulting function from A to B ...
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1answer
35 views

Compact embedding

Prove that the embedding $j\colon (C^1[0,1],\|\cdot\|)\to(L^1[0,1],\|\cdot\|_{L^1})$ where $\|f\|=\max\{\|f\|_\infty,\|f'\|_\infty\}$ and $\|f\|_\infty$ denotes the supremum norm, ...
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0answers
35 views

Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
2
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2answers
193 views

How to complete this proof to show that the metric $d'(x,y) = d(x,y) / (1 + d(x,y))$ gives the same topology as $d(x,y)$ gives?

This is an exercise problem from Munkres's Topology (Exercise 11 of Section 20 "The Metric Topology", 2nd edition). Exercise 11: Show that if $d$ is a metric for $X$, then $$d'(x,y) = d(x,y) / (1 ...
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1answer
52 views

Proving via axioms, that for given set $A$, $P(P(A))$ exists

The question itself: For a given set A, prove P(P(A)) exists. You may only use the axiom of pairing, axiom of union and axiom of empty set. This is how I solved it: Let A be the given set. ...
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1answer
159 views

Direct combinatorial proof of a sum identity on formal Lagrange polynomials

Let $k$ be a field and $K=k(x_0,x_1,\ldots, x_n)[x]$. Define $$\mathcal{L}_k(x)\triangleq \prod_{\substack{j=0\\ j\ne k}}^n\frac{x-x_j}{x_k-x_j}.$$ Is there a purely combinatorial way to show ...
2
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2answers
123 views

The center of a group with order $p^2$ is not trivial

Let $p$ be a prime and $G$ be a group of order $p^2$. Show that $Z(G)\neq 1$. Is there a proof of this nice fact that doesn't use the class equation?
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0answers
55 views

The Toad and Frog Game - Proof by Inducation

Toads and Frogs is played on a 1 × n strip of squares. At any time, each square is either empty or occupied by a single toad or frog. Although the game may start at any configuration, it is customary ...
16
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2answers
786 views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as simply as possible

Can someone prove the special case of Fermat's Last Theorem for $n=3$, i.e., that $$x^3 + y^3 = z^3,$$ has no positive integer solutions, as simply as possible? I have seen some good proofs, but ...
2
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1answer
80 views

How to show that $\lfloor n/1\rfloor+\lfloor n/2 \rfloor+…+\lfloor n/n\rfloor+\lfloor{\sqrt{n}}\rfloor$ is even?

Let $n$ is a natural number. Prove that $$\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+....+\left\lfloor\frac{n}{n}\right\rfloor+\left\lfloor{\sqrt{n}}\right\rfloor$$ ...
2
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0answers
82 views

The Heine-Borel Theorem for the real line

Hi everyone I'd like to know if the following argument is correct and also I'm very interested in a constructive approach for (2)$\Rightarrow$(1) (a link or a hint it will sufficient for me) I was ...
0
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2answers
87 views

Matrix power series has linearly dependent terms

Prove that for any $(n\times n)$ real matrix, the set of matrices $\{I,M,M^2,...,M^n\}$ are linearly dependent. More formally, we have to prove that $$\forall M \in \mathbb{R}^{n \times n},\\ ...
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0answers
332 views

Easier Solution? - Find plane perpendicular to another plane and through the intersection line of two planes [Stewart P803 12.5.38]

$38.$ Find an equation of the plane that's $\perp$ the plane $x + y - 2z = 1$ and passes through the line of intersection of the planes $x - z = 1$ and $y + 2z = 3$. $\bbox[3px,border:2px solid ...
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1answer
165 views

Problem with alternate solution — Equation of plane through point and containing intersection line of two planes [Stewart P $803, 12.5.37$]

$37.$ Find an equation of the plane that passes through the point $(1, -2, 1)$ and contains the line of intersection of the planes $x + y - z = 2$ and $2x - y + 3z = 1$. $\bbox[3px,border:2px solid ...
1
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0answers
71 views

How to prove that the inverse of a persymmetric matrix is also persymmetric?

An exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...
3
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0answers
43 views

Using an automatic tool for checking geometric conjectures

I do a lot of research about squares, and I thought of using some automatic tool for proving / disproving some geometric conjectures. As a simple example, consider the following Square coloring ...
3
votes
1answer
128 views

An alternate analysis to the (worst-case) run time of the euclidean algorithm

I was trying to figure out the running time of the euclidean algorithm. The analysis that I found on Wikipedia and CLRS both analyze the run time of the euclidean algorithm using the Fibonacci ...
7
votes
1answer
315 views

Uniform Convergence verification for Sequence of functions - NBHM

Following is a list of problems from an exam for admission into Ph.D program. I have just compiled all previous questions on uniform convergence of sequence of functions and i tried to work out . I ...