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

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3
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1answer
74 views

On Elements of $p$th Row in n Pascal's Triangle (For Prime $p$)

If $p$ is a prime number, in Pascal's triangle all the terms in the $p$th row - except the 1s - are multiples of $p$ . It's easy to prove this property using the formula for $\binom{p}{k}$. Is there ...
1
vote
0answers
36 views

Ineqality regarding LCM of $1, 2, \ldots, n$

While going through F. Beukers proof of irrationality of $\zeta(3)$ I found the inequality $d_{n} < 3^{n}$ for all sufficiently large values of $n$ where $d_{n}$ denotes the LCM of all the numbers ...
1
vote
0answers
58 views

Maximally Consistent Set (Proof by Contradiction)

Yesterday, I asked about feedback for a proof of the following theorem For all $\phi$, $\phi \in \Gamma^{*}$ if and only if $\Gamma^{*} \vdash \phi$. My main concern was the first part $(\to)$, which ...
0
votes
0answers
39 views

On Maschke's theorem: proof with matrix skills

In page 4 by Isaacs' book "character theory of finite groups", there is a proof for Maschke's theorem. In page 48 by Curtis and Reiner's book Representation theory of finite groups and associative ...
1
vote
4answers
135 views

Elegant or elementary evaluation of $\lim\limits_{x\to 0} \left( \frac{1}{x}-\frac{1}{\sin(x)} \right) $ [duplicate]

I give math tutoring and was wondering about the following limit. I found the answer but I was wondering if someone has a nicer explanation than the one I am giving where I use L'Hôpital's rule twice. ...
0
votes
2answers
67 views

prove there exists $x$ in ${\mathbb R}$ using the completeness axioms

Let $a, b \in {\mathbb R}$ with $a < b$. Prove that there exists $x\in {\mathbb R}$ which is NOT a rational number such that $a < x < b$. This is what i have at the moment.. It does not have ...
4
votes
3answers
154 views

Alternative Creative Proofs that $A_4$ has no subgroups of order 6

Since I've been so immersed in group theory this semester, I have decided to focus on a certain curious fact: $A_4$ has no subgroups of order $6$. While I know how to prove this statement, I am ...
0
votes
1answer
105 views

Showing that $a_n \not \to 17$ implies a subsequence $a_{n_k}$ that is $\epsilon$ far from $17$ for some $\epsilon > 0$

I want to check my proof for this question: Suppose a sequence {$a_n$} does not converge to 17. Prove that there exists some $\epsilon$ > 0 and a subsequence {$a_{nk}$} so that $|a_{nk} - 17|$ > ...
2
votes
1answer
119 views

Help with a lemma of the nth root (without the binomial formula)

I have no idea of how to solve it. I would appreciate if someone gives me a hint, please. Definitions Let $\,x^{1/n}:= sup\{\, y \in \mathbb{R}: y\ge0 \text{ and } y^n\le x\, \}$ Lemma: Let ...
2
votes
2answers
193 views

Verification of proof of the Sequence of Arithmetic Theorem

Suppose $\left\{b_{n}\right\}$ is a sequence of real numbers which converges to $M$, so that $b_{n} \neq 0$ for each $n$, and $M \neq 0$. Prove that the sequence $\{ \frac{1}{b_n} \}$ converges to ...
0
votes
2answers
111 views

$n!\sum_{k=1}^n \frac{a_{k}}{k!}$ is always integer.

$\displaystyle n!\sum_{k=1}^n \frac{a_{k}}{k!}\in \mathbb{Z}$ where $n,a_k$ and $k$ are integers. I know the proof by induction. Is there any other technique to prove it? Thank you.
5
votes
1answer
160 views

Alternative argument - set theory problem

I tried to solve the following problem: Let $ \mathcal{F}$ be a nonempty family of sets with the following properties: (a) If $ X \in \mathcal{F}$, then there are some $ Y \in \mathcal{F}$ ...
1
vote
2answers
1k views

Prove that $\sup(A+B) = \sup(A) + \sup(B)$ and why does $\sup(A+B)$ exist?

We want to show that $\sup(A)+\sup(B)$ is the least upper bound of the set $A + B$. First, we need to show that $\sup(A) + \sup(B)$ is an upper bound for the set $A + B$. Indeed, if $z\in A + B$, then ...
4
votes
4answers
181 views

Prove $3^n \ge n^3$ by induction

Yep, prove $3^n \ge n^3$, $n \in \mathbb{N}$. I can do this myself, but can't figure out any kind of "beautiful" way to do it. The way I do it is: Assume $3^n \ge n^3$ Now, $(n+1)^3 = n^3 + ...
2
votes
1answer
211 views

Proof for $-\sup(A) = \inf(-A)$

Let $A$ and $-A = \{ -x \mid x \in A \}$ be two bounded sets. I have to prove that $-\sup(A) = \inf(-A)$, i did it in the following way and wish to know if it is sufficient: $ \exists x\in A$ such ...
5
votes
1answer
196 views

Dividing Squares Fails to Invoke Contradiction: Two Elementary Divisibility Proofs

$x^2 \text{ is even } \iff x \text{ is even } \tag{Thm 3.12, P76}$ $\text{ Let } x, y \in \mathbb{Z}. \text{ Then } x \;\& \; y \text{ are of the same parity } \iff x + y \text{ is even.} \tag{Thm ...
2
votes
1answer
114 views

Every finite group is finitely generated. - alternate proof

Is that's all? Thank you. :-) A group $H$ is called finitely generated if there is a finite set $A$ such that $H = \left \langle A \right \rangle$ . Prove that every finite group is finitely ...
5
votes
0answers
127 views

Higman group with 3 generators is trivial

The group $G$ generated by $x,y,z$ subject to the relations $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ is trivial. This isn't the case for the corresponding group with 4 generators, which is the famous Higman ...
0
votes
2answers
125 views

Prove $\log_a(b)$ is irrational given that $a, b$ are positive distinct primes.

I know this is a classical proof by contradiction exercise, and there are full solutions else where, doing a quick search I didn't find any, but I would approach this question like this: Suppose ...
5
votes
0answers
97 views

Is there some elementary proof of invariance of domain?

Invariance of domain at least in statement seems a simple result. I mean, the first time I saw the statement I thought: "the proof can't be that bad", but when I searched for it I saw that it needs ...
2
votes
3answers
689 views

Prove that in a parabola the tangent at one end of a focal chord is parallel to the normal at the other end.

Prove that in a parabola the tangent at one end of a focal chord is parallel to the normal at the other end. Now, I know prove this algebraically, and that's very easy, but I am not getting any ...
1
vote
1answer
81 views

How to improve this proof about real numbers?

I've trying to prove the following: let $a \in \mathbb{R}$ with $a \geq 0$ if for every real $\epsilon > 0$ we have $0 \leq a < \epsilon$ then $a =0$. I'm sure that there is a proof much ...
0
votes
0answers
79 views

Elementary proof of Hopf's theorem

I am looking for a proof of the following theorem: If we have $n$-dimensional division algebra, then $n=2^k$ for a $k\in \mathbb{N}_0$. Both proofs that I have seen are based on the same idea ...
1
vote
1answer
53 views

If $p$ is irreducible and $p \not \mid a$, then $\text{gcd}(p,a)=\pm 1$.

I will be taking a Rings and Fields course in the Fall, so I figured I would read ahead in the textbook (A First Course in Abstract Algebra, by Anderson and Feil) to prepare. Recall the following ...
3
votes
2answers
212 views

Fundamental theorem of algebra: a proof for undergrads?

The fundamental theorem of algebra is the statement that a complex polynomial of positive degree has at least one root. I do not know complex analysis but I searched for proofs of the statement and ...
3
votes
5answers
175 views

$\sum_{k=1}^nH_k = (n+1)H_n-n$. Why?

This is motivated by my answer to this question. The Wikipedia entry on harmonic numbers gives the following identity: $$ \sum_{k=1}^nH_k=(n+1)H_n-n $$ Why is this? Note that I don't just ...
5
votes
5answers
163 views

interesting Integral , alternative solution.

Show the following relation: $$\int_{0}^{\infty} \frac{x^{29}}{(5x^2+49)^{17}} \,\mathrm dx = \frac{14!}{2\cdot 49^2 \cdot 5^{15 }\cdot 16!}.$$ I came across this intgeral on a physics forum and ...
5
votes
2answers
266 views

Is my proof correct about limit of $\sin\left(\frac{1}{x}\right)$?

Apostol's book Calculus asks to show that there is not a value $A$ such that $f(x)=\sin\left(\frac{1}{x}\right)\to A$ when $x \to 0$. And my proof is: Suppose for the sake of contradiction that there ...
2
votes
3answers
414 views

Axiom of Choice and Right Inverse

I read an Theorem that states: Let $A$ and $B$ be non-empty sets, and let $f:A \to B$ be a function, then the function $f$ has a right inverse if and only if $f$ is surjective. The Theorem ...
6
votes
11answers
783 views

How to prove that $\lim\limits_{x\to0}\frac{\tan x}x=1$?

How to prove that $$\lim\limits_{x\to0}\frac{\tan x}x=1?$$ I'm looking for a method beside L'Hospital's rule.
1
vote
1answer
152 views

Alternative proof or verification of given proof of convergence in probability

I am asked to show that if $X_n \rightarrow c$ in probability and if $g$ is a continuous function, then $g(X_n) \rightarrow g(c)$ in probability for a statistics homework problem in a section titled ...
1
vote
1answer
102 views

Identity concerning complete elliptic integrals

It can be easily checked that both the complete elliptic integrals $K(k), K'(k)$ satisfy the same second order differential equation $$kk'^{2}\frac{d^{2}y}{dk^{2}} + (1 - 3k^{2})\frac{dy}{dk} - ky = ...
1
vote
1answer
62 views

proving inequality $0 < x^4+2x^2-2x+1$ for $x>0$

How can I elegantly prove the inequality $0 < x^4+2x^2-2x+1$ for $x>0$. I have plotted this function in a Sage (an open source and free CAS) and I can see that there is a local min between $0$ ...
16
votes
5answers
476 views

A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$

This is a problem from "A Course of Pure Mathematics" by G H Hardy. Find the limit $$\lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$$ I had solved it long back (solution presented in my blog ...
1
vote
4answers
2k views

Prove that a continuous function on a closed interval attains a maximum

As the title indicates, I'd like to prove the following: If $f:\mathbb R\to\mathbb R$ is a continuous function on $[a,b]$, then $f$ attains its maximum. Now, I do have a working proof: $[a,b]$ ...
0
votes
3answers
43 views

If $G$ is disconnected and the vertices $x,y$ are adjacent in $G$, then there is a vertex that isn't adjacent to $x$ and isn't adjacent to $y$.

I'm just starting graph theory and I'm trying to prove the following: Let $G$ be a simple disconnected graph with vertex set $V(G)$ and edge set $E(G)$. If $x,y\in V(G)$ and $xy \in E(G)$, then ...
1
vote
1answer
47 views

Prove A and B are congruent

Prove A and B are congruent where $i_1i_2\text{...}i_n$ is a permutation of $1,2,\text{...},n$ My proof. Swapping two elements in the diagonal of Matrix A is identical to swapping two rows, ...
1
vote
0answers
53 views

Prove: swapping 2 adjacent numbers will change the parity of the permutation

Prove: swapping 2 adjacent numbers will change the parity of the permutation proof: Suppose the permutation of n numbers is as following,with arbitrary adjacent two numbers are $p_i$ and ...
1
vote
1answer
204 views

Element-wise proofs vs. direct proofs from definitions

First off, this post is a question; I hope to get constructive feedback on this proof. I would not consider anything below correct, it just makes sense to me. I am trying to learn Real Analysis and ...
2
votes
1answer
221 views

questions about Rudin's summation by parts

rudin's summation by parts 3.41 Theorem Given two sequences $\left\{a_n\right\}$,$\left\{b_n\right\}$,put $$\begin{align*}A_n=\sum _{k=0}^n a_k~~~(1)\end{align*}$$ if $n\geq 0$; put $A_{-1}=0$. ...
7
votes
11answers
811 views

limit question: $\lim\limits_{n\to \infty } \frac{n}{2^n}=0$

$$ \lim_{n\to\infty}\frac n{2^n}=0. $$ I know how to prove it by using the trick, $2^n=(1+1)^n=1+n+\frac{n(n-1)}{2}+\text{...}$ But how to prove it without using this?
3
votes
4answers
141 views

Prove $\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A$ by the Pythagorean theorem.

How do I use the Pythagorean Theorem to prove that $$\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A?$$
9
votes
0answers
218 views

Pólya and Szegő, Part I, Ch. 4, 174.

The following is a problem proposed in Pólya and Szegő's book "Problems and Theorems in Analysis" Assume that $0<f(x)<x$ and $$f(x)=x-ax^k+bx^\ell+x^\ell \varepsilon(x),\,\;\;\;\lim_{x\to ...
6
votes
0answers
202 views

A few standard results (on metrizability and relative separation strength) without choice?

I've been going back over some results from Munkres's Topology, and I'm curious about some things.... I know that Choice principles have some connection to the separation axioms (in ZF, at ...
1
vote
1answer
84 views

Proofs that $f^{(100)}(0)= -\frac1{101}$, where $f(x) = \frac{\sin x}x$ and $f(0)=1$

4, Let $$f(x)=\begin{cases} \dfrac{\sin{x}}{x}&x\neq 0\\ 1&x=0 \end{cases}$$ Find the value $f^{(100)}(0)$. I find that $$f^{(100)}(0)=-\dfrac{1}{101}$$ by noticing ...
9
votes
3answers
128 views

Please review my question and solution. Thanks in advance.

How many values of x are there such that there exists positive integer solutions for S, such that $S=\sqrt{x(x+p)}$ where $x$ is an integer and $p$ is a prime number $>2$ This is a problem I made ...
4
votes
3answers
267 views

Why are only fractions with denominator 2 and 5 non-repeating?

Given a rational number $\frac{n}{d}$, I understand that in the base $10$ number system, the number can be represented as a non-repeating decimal number if and only if $d$ has only prime factors of ...
3
votes
1answer
85 views

Prove: if $f(x) =x\sin (\pi x)$ then $f'(x)$ vanishes at a unique point in $ ( n + 1/2, n) $

Let $ f(x) = x\sin (\pi x), x > 0 $. Then prove that for all natural numbers n, $f'(x)$ vanishes at a unique point in $ ( n + 1/2, n) $ The given solution shows a graph, but is there any algebraic ...
6
votes
1answer
149 views

Combinatorial proof of the fact $p$ doesn't divide $ n \choose p^k$

Let $p^k | n$ and $p^{k+1} \nmid n$. Is there any combinatorial proof of the fact that $p \nmid {n \choose p^{k}} $ ?
5
votes
1answer
217 views

Using Nakayama's Lemma to prove isomorphism theorem for finitely generated free modules

Suppose $R \neq 0$ is a commutative ring with $1$. The following is well known: (Isomorphism Theorem for Finitely Generated Free Modules) [FGFM] $R^{n}\cong R^{m}$ as $R$-modules if and only if ...