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

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37 views

Prove that $\dfrac{\sigma_1(n)}{n} = \sigma_{-1}(n)$ where $\sigma_x(n)$ is the sum of the $x$th powers of the positive divisors of $n$.

I computed $\dfrac{\sigma_1(n)}{n}$ and $\sigma_{-1}(n)$ on a good hundred values of $n$, and they seem to always match. For example: $\dfrac{\sigma_1(6)}{6} = \dfrac{1 + 2 + 3 + 6}{6} = ...
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2answers
62 views

Prove that n!+1 contains a prime factor greater than n and use this to prove that there are infinte many primes [duplicate]

Prove that $n!+1$ contains a prime factor greater than $n$ and use this to prove that there are infinitely many primes. I said assume that $n!+1$ contains a prime $p$ which is less than or equal to ...
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1answer
37 views

Convergence of Alternating harmonic series (Direct!)

Once again, note No use of the alternating series test!
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1answer
42 views

Left & right adjoints in the context of complete lattices.

This is a follow-up question from this question of mine. In the same paper as the one mentioned in my previous post, it's stated that In the context of complete lattices, a monotone map has a ...
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2answers
100 views

inequality $\prod\limits_{k=1}^n\frac{2k-1}{2k}\lt\frac1{\sqrt{3n}}$

$n$ is a positive integer, then $$\prod_{k=1}^n\frac{2k-1}{2k}\lt\frac1{\sqrt{3n}}$$ with mathematical induction, we can prove this. But I would love to find a wonderful method without ...
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1answer
44 views

Orbit-Stabiliser Theorem Application

Question Let $G$ be the symmetric group $S_n$ acting on the $n$ points $\lbrace 1, 2, 3, . . . , n\rbrace$, let $g\in S_n$ be the n-cycle $(1,2,3,. . . , n)$. By applying the Orbit-Stabiliser Theorem ...
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1answer
107 views

Seeking a more direct proof for: $m+n\mid f(m)+f(n)\implies m-n\mid f(m)-f(n)$

If $f:\mathbb N\to\mathbb Z$ satisfies: $$\forall n,m\in\mathbb N\,, n+m\mid f(n)+f(m)$$ How to show that this implies: $$\forall n,m\in\mathbb N,\,n-m\mid f(n)-f(m)?$$ I was almost incidentally ...
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1answer
83 views

Prove that the dual graph of any (planar) graph is connected

I'd like to know if there's a standard proof that the dual graph of any planar graph is connected (or, if there's a counterexample, I'd like to know that too). I've thought of a proof that might work ...
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4answers
172 views

$e^{i\theta}$ $=$ $\cos \theta + i \sin \theta$, a definition or theorem?

My question is simply whether the well-known formula $e^{i \theta}$ $=$ $\cos \theta$ $+$ $i \sin \theta$ a definition or there is some proof of the result. It seems to me that the formula is a ...
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1answer
46 views

Validity of “circular” proofs

I believe this is an easy question. I put circular in quotations because I'm pretty sure I'm not talking about circular proofs in general. I was thinking about how to prove that any function whose ...
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3answers
91 views

Proving the open interval $(0,1)$ is uncountable [duplicate]

I am currently able to prove this statement using the Cantor diagonalisation argument, my question is whether there is another way (more simple or more complex) to prove this statement, without ...
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3answers
758 views

**Ended Competition:** What is the shortest proof of $\exists x \forall y (D(x) \to D(y)) $?

The competition has ended 6 june 2014 22:00 GMT The winner is Bryan Well done ! When I was rereading the proof of the drinkers paradox (see Proof of Drinker paradox I realised that $\exists x ...
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1answer
41 views

Same characteristic polynomial $\iff$ same eigenvalues?

This proves: Similar matrices have the same characteristic polynomial. (Lay P277 Theorem 4) I prefer http://math.stackexchange.com/a/8407/53259, but this proves that they have the same eigenvalues. ...
4
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1answer
102 views

Characterization of positive definite matrix with principal minors

A matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, for ...
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2answers
355 views

A snappy proof of Fatou's lemma

I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ...
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0answers
26 views

A compendium of proof-techniques per objective

Please consider this as an on-going list of techniques preferably per objective or subject. Many mathematical books (at least lately) are focusing on "design patterns" if you like of proof-techniques ...
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2answers
30 views

Different way showing a subgroup is a subgroup of another subgroup

http://crazyproject.wordpress.com/2010/04/11/subgroups-and-quotient-groups-of-solvable-groups-are-solvable/ Lemma 1: Let $G$ be a group and let $H,K,N \leq G$ with $N$ normal in $H$. Then $N \cap K$ ...
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1answer
59 views

Force between two parallel wires?

Having two current carrying (currents $I'$ and $I$) wires of length $a$ parallel to the $z$-axis, one with end points $(0,0,0)$ and $(0,0,a)$ and one from $(a,0,0)$ to $(a,0,a)$, I'm looking for the ...
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0answers
54 views

Find a short formula for $\sin x+\sin (x+y)+\sin (x+2y)+. . .+\sin (x+(n-1)y)$

The answer is : $$\sin(\frac {x+x+(n-1)y}{2}) \dfrac {\sin \frac{ny}{2}}{\sin \frac {y}{2}}$$ I could've written the question as: Show that..., but then people would try induction. What I did: ...
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1answer
56 views

Can some one explain me a easy alternate proof of rank nullity theorem?

OK I have just gone through the Gilbert Strang's 'Introduction to Linear algebra'. I felt it as a very nice book. But I am bad at proving things. I have an idea what null space (kernel) and column ...
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0answers
89 views

Fourier Series of $f(x)=e^x$ on $[0,\pi)$ as a function of period $\pi$

Can you tell me what you get? I've tried computing it, I've got some result but I don't think it's right since I need to use it for something else and it doesn't work at all... What exactly I'm trying ...
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5answers
179 views

Prove that $4^n$ is not divisible by 3.

How can one prove that $4^n$ is not divisible by 3, for any $n \ge 0$? One way I found is to proof that $4^n - 1$ is always divisible by 3 (as demonstrated in a question here), thus $4^n$ could ...
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0answers
36 views

Proving a simple claim concerning order without using LaGrange's Theorem

For whatever reason, I am having trouble proving the following claim without using LaGrange's Theorem. Claim: Let $G$ be a group of order $n < \infty$. Then, $x^{n}=1$, where $1$ is the identity, ...
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1answer
84 views

Prove that if $S$ has a greatest element $b$, then $b = lubS$.

Prove that if $S$ has a greatest element $b$, then $b = lubS$. These are the definitions I used: Def. Given a partially ordered set ($P, \le$), then an element $b$ of a subset $S$ of $P$ is the ...
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1answer
81 views

A Shorter Proof of Rosser's Theorem Without Using The Prime Number Theorem

While researching on the elementary proof of Bertrand's Postulate I came to know about a theorem of Rosser's which states that $p_n$ $>$ $n$ $\text{ln}$ $n$. I have seen Rosser's original proof and ...
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2answers
36 views

Easier Proof - Union of finite lin-indep subsets of the eigenspaces = a lin-indep subset. [Lay P285 Thm 5.3.7c]

P267 Lemma. Let $T$ be a linear opera $tor$, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. For eacb $i=1,2,\ \ldots,\ k$, let $v_{i}\in E_{\lambda;}$, the ...
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1answer
38 views

Adaptation of this proof of spectral theorem to the complex case

My question is quite simple, I would like to know why we can't use this proof to the complex case, i.e., the operator $T$ is self adjoint on a complex n-dimensional inner product space $V$. Can we ...
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2answers
35 views

Geometric proof and extension of |a|=|b|=|c|=a+b+c=1 => a=1 or b=1 or c=1

We have $a,b,c\in\mathbb{C}$ verifying $|a|=|b|=|c|=a+b+c=1$, we have to show that $a=1$ or $b=1$ or $c=1$. That can be rather easily proved using trigonometry formulas. Is there a way to prove it ...
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0answers
45 views

Mathematical logic and proofs involving absolute values

Is the following proof correct? Let ($\forall$ x, y $\in $$\Bbb R)$ $|x-y| \le |x| +|y|$ case#1: Suppose x and y $\ge0$. We want to show that $|x-y| \le |x| + |y|$. Since $|x|\ge x$ and $|y|\ge y$, ...
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1answer
55 views

Is convex set in an ordered set necessarily interval or ray? Munkres 16. 7

The is Problem 7 in Section 16 (page 92) of Munkres' Topology. The problem reads as follows. Let $X$ be an ordered set. If $Y$ is a proper subset of $X$ that is convex in $X$, does it follow ...
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0answers
28 views

3-connectedness Expansion Lemma

The question is as follows: Prove that applying the expansion operation* to a 3-connected graph yields a 3-connected graph. *The expansion operation is as follows: you take two edges of a graph ...
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0answers
82 views

A power of the characteristic polynomial

Let $A$ be a square matrix with real or complex coefficients of size $n$. Define its characteristic polynomial by $\chi_A(X) = \det(A-XI_n)$ (or $\det(XI_n-A)$ if you prefer). The question is : Prove ...
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3answers
1k views

Show that the product of two consecutive natural numbers is never a square.

I'd like to have my proof verified and if possible, to see other solutions that are interesting. Proof: Suppose $n(n+1)$ is a square. Then we write $$n(n+1) = \prod_{p} p^{c(p)}$$ where $c(p) = a(p) ...
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2answers
91 views

Proof for $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ without complexes? [duplicate]

This is what I needed. Practically, a link were also okay. $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$
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3answers
246 views

Proving $fg$ and $f+g$ is Riemann integrable through the easy and hard way.

Problem: Suppose $f,g$ are Riemann integrable functions, show that $f+g$ and $fg$ are also Riemann integrable. I know there is really easy to do this with measure theory, but I want to see if ...
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2answers
35 views

Simple proof of “$a$ and $a^{-1}$ have the same number of conjugates”

I recently had to give a proof of this, I gave a correct proof but I feel that it was overly complicated, so the question here is "Find a simpler proof (one that belongs in "the book")". Question: ...
2
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2answers
117 views

Solve $ax \equiv b \mod m$ without Linear Congruence Theorem or Euclid's Algorithm?

Origin page 5. The overhead doesn't look like Linear Congruence Theorem or anything from Euclid's Algorithm. page 4 tries to delineate ...
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0answers
28 views

Alternative proof of Cauchy's integral formula on the polydisc in several variables

I've been reading a few different texts on several variable complex analysis and each proof I've seen of Cauchy's integral formula (on the polydisc) has effectively been an inductive proof. My ...
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1answer
87 views

Prove $p$$_n$$_+$$_1$ $<$ $2p_n$ without using the Bertrand's Postulate [closed]

Recently I have been researching on the Bertrand's Postulate to find and elementary proof of it. I have been able to prove that (if I have not made a very pathetic mistake) for any composite $n$ ...
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0answers
45 views

bounding the sum of squares of lengths of a quadrilateral inscribed in a unit square

Consider this nice little problem: if $ABCD$ is a quadrilateral inscribed in a unit square, then $$2\leq AB^2+BC^2+CD^2+DA^2\leq4$$ (Evidently this is problem 1 on paper 1 of the 1989 Irish ...
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2answers
119 views

Prove that if $A$ is normal, then eigenvectors corresponding to distinct eigenvalues are necessarily orthogonal (alternative proof)

The problem statement is as follows: Prove that for a normal matrix $A$, eigenvectors corresponding to different eigenvalues are necessarily orthogonal. I can certainly prove that this is the ...
3
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0answers
86 views

Heron's Formula; an Intuitive or Visual Proof

I've found several proofs for Heron's formula for the area of a triangle in term of its sides, but none of them is simple and intuitive enough to show WHY the formula works. Do you know an intuitive ...
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3answers
62 views

Sequence that contains subsequences converging to every point in the infinite set $\{1/n \} \, \forall \, n \in N$ (Abbott p 58 q2.5.3c)

has this property. Notice that there is also a subsequence converging to 0. We shall see that this is unavoidable. I acquiesce to this example, but I wasn't conscious of it until I read ...
3
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4answers
72 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
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1answer
44 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 ...
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0answers
27 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
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1answer
65 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 $$ ...
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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 ...
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1answer
73 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
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1answer
26 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 ...