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

learn more… | top users | synonyms

3
votes
4answers
76 views

A more rigorous way to prove this? [duplicate]

I would like to prove the following statement $$x^n-a^n=(x-a)\sum^{n-1}_{k=0}x^ka^{n-k-1},\qquad\forall n\in\Bbb N_0$$ I can easily prove it by induction using polynomial long division or series ...
3
votes
1answer
65 views

A Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues - Proof Strategy [Lay P397 Thm 7.1.3c]

Would someone please explain the proof strategy at Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues? I brook the algebra so I'm not asking about ...
2
votes
0answers
34 views

Proving $d_E$ is a Metric on $\Bbb R^n$

This question rather then wanting help w.r.t (M3) (The Triangle Inequality ) with can be done by using the Cauchy–Schwarz inequality. I would like advice regarding (M1): $d_E(\mathbf {q,p}) = 0 \iff ...
3
votes
1answer
83 views

Determinant of a matrix with symmetric positive definite block

In reviewing linear algebra for an exam, I encountered the following problem: Let $A \in \mathbb{R}^{n\times n}$ be symmetric positive definite. If $x$ is any nonzero vector, show that $$ ...
1
vote
0answers
25 views

Show that $B(X)$ is semisimple for a Banach space $X$ [duplicate]

Show that $B(X)$ is a semisimple Banach algebra, where $X$ is a Banach space. That is, to show that rad $B(X)=\{0\}$, or equivalently, to show $\sigma(AT)={0} \, \forall T\in B(X)\Rightarrow A=0$. I ...
0
votes
1answer
81 views

Inequality proof, why isn't squaring by both sides permissible?

Suppose $a$ and $b$ are real numbers. Prove that if $0 < a < b$ then $a^2 < b^2$. I understand that the normal way to prove this is to multiply $a < b$ by $a$ and then by $b$ and then ...
0
votes
1answer
36 views

Tiling an $m\times n$ grid.

For natural numbers $m$ and $n$, an $m\times n$ grid of squares can be tiled with tiles of the form completely filling the grid, without overlapping, if and only if $m,n\geq2$ and $6\mid mn$. It ...
6
votes
1answer
142 views

Equivalence of definitions for “normal extension” and how to lift isomorphisms to them

Briefly: I want to prove that these two definitions for "normal extension" are equivalent: "$K$ is a splitting field for a collection of polynomials in $F[x]$" vs. "Every irreducible polynomial in ...
0
votes
1answer
76 views

$x_n,x_ny_n$ convergence implies $y_n$ converges

Assume that $x_n$ converges to a nonzero number $x$ and that the sum $x_ny_n$ converges to a limit $L$. Prove that the series $y_n$ converges. The natural guess is that $y_n$ will converge to $L/x$. ...
2
votes
1answer
124 views

Valid Proof for Cayley Hamilton Theorem? (Not the usual incorrect one)

By induction; case n=1 is true. $A$ admits an eigenvalue $\lambda$ with eigenvector $v$ over $\mathbb{C}$. Change $A$ into a basis $e_1=v,...,e_n$. Then $\exists X$ such that $XAX^{-1}=\left( ...
0
votes
0answers
111 views

indepence transitive property?

For the events A and B are independent and B and C are independent is A and C independent I used coin tosses to try to model this with A = H B = T and C = H in seperate fair tosses I get that they ...
4
votes
4answers
237 views

Prove that these two curves have the same length

My midterms are approaching, and I was going through some of our past Calculus midterms when I stumbled upon this question from 1996: Show that these two curves, $$(\Gamma) : \frac ...
1
vote
2answers
73 views

Is there an elementary proof that $n \mid \phi(p^n-1)$?

When I teach finite fields, one fun corollary is that $n \mid \phi(p^n-1)$, where $\phi$ is the Euler-phi function and $p$ is prime. I spent several minutes in my office after class one day looking ...
0
votes
3answers
48 views

Proof that $X^C \cap Y^C= \;(X \cup Y)^c$

Proof that if $X \subset S,\; Y\subset S,\;$ then $\;X^C \cap Y^C= \;(X \cup Y)^c:$ It must be shown that the two sets have the same elements, that each element of the set on the left is an element ...
3
votes
3answers
88 views

Could this linear algebra proof be done without computation?

From page 95 of Hoffman & Kunze's Linear algebra: Let $T$ be the linear operator on $\mathbb{R}^2$ defined by $T(x_1,x_2)=(-x_2,x_1)$ Prove that if $B$ is any ordered basis ...
3
votes
0answers
80 views

Banach Alaoglu different proofs

While trying to prove Banach-Alaoglu theorem I noticed the differrent equivalent definitions of compactness. When I tried to find a proof of Banach-Alaoglu I found a proof in Pedersen Analysis Now and ...
0
votes
2answers
60 views

Direct understanding of vector projection

I'm aware that we can project an arbitrary vector $v$ onto a unit vector $\hat{\mathbf{k}}$ by $(\mathbf{v} \cdot \hat{\mathbf{k}})\hat{\mathbf{k}}$ But why is this true? I would imagine that such ...
5
votes
2answers
114 views

Cool property of the number $24$

Recently I've had my 24th birthday, and a friend commented that it was a very boring number, going from 23 which is prime, 25 which is the first number that can be written as the sum of 2 different ...
2
votes
3answers
30 views

Can someone explain me how to read members of a set to prove uncountability?

I have trouble in understanding what the elements in the following set are: $$ V = \{f:\mathbb{N}\to \mathbb{N} \mid \text{there is $N_f \in \mathbb{N}$ so that $f(x) \le N_f$ for all ...
1
vote
3answers
69 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
votes
1answer
38 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 ...
0
votes
2answers
55 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
votes
0answers
59 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 ...
1
vote
5answers
72 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 ...
1
vote
1answer
54 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} ...
1
vote
1answer
37 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 ...
0
votes
0answers
37 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
votes
2answers
85 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: $$(1\mod{10})^2 = 1\mod{10}$$ $$(2\mod{10})^2 = 4\mod{10}$$ ...
0
votes
2answers
181 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 ...
0
votes
2answers
54 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 = ...
1
vote
2answers
50 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 ...
13
votes
4answers
340 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 ...
0
votes
0answers
49 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 ...
0
votes
1answer
99 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
votes
3answers
120 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 ...
1
vote
6answers
255 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. How can we show that the only solution to this equality is $a'=na$ and $b'=nb$, $n$ is a natural ...
4
votes
2answers
336 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
votes
0answers
55 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
votes
2answers
83 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 ...
0
votes
1answer
78 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 ...
1
vote
1answer
64 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 $<A_1A_2A_3\cdots A_n>$, ...
3
votes
0answers
169 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
vote
1answer
77 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
votes
1answer
302 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 ...
1
vote
1answer
90 views

Find the coefficients such that all four roots of $(x^2-px+q)(x^2-qx+p)$ are natural numbers

Find all ordered pairs $(p,q)$ of natural numbers such that all $4$ the roots of $$f(x)=(x^2-px+q)(x^2-qx+p)$$ are natural numbers. I got a solution of the problem (see below) but I want some ...
21
votes
9answers
776 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
votes
2answers
74 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 ...
1
vote
0answers
31 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
votes
2answers
103 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}} ...
10
votes
4answers
373 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 that it can be proved using Weierstrass Theorem, ...