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

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2
votes
1answer
102 views

Check proof please

Prove that if: $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ than: $\lim_{x\rightarrow\infty}{\frac{f(x)}{x}}=L$ Assuming $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ we can choose $X_{\epsilon}$ s.t. ...
4
votes
2answers
156 views

Proof about AM-GM inequality generalized

Note: I'm not sure this type of questions are welcome on the site. In case tell me. Let's define the $p$ mean as $$M_p(x_1, \dots, x_n) = \sqrt[p] { \frac 1n \sum_{i = 1}^n x_i^p}$$ for $x_1, ...
2
votes
3answers
98 views

If $I_{n+1}\subset I_n$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty

Question: If $I_n$ is closed and bounded, $I_{n+1}\subset I_n$, and $I_n\neq\emptyset$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty. This is not a homework help question. I'm actually looking ...
5
votes
1answer
80 views

All $k$-regular subgraphs of $K_{n,n}$ have a perfect matching: a proof without Hall's Marriage Theorem?

There are several ways of describing this result: Theorem: For $k \in \{1,2,\ldots,n\}$, any $k$-regular subgraph of $K_{n,n}$ has a perfect matching (also known as a $1$-factor). I tend to ...
2
votes
1answer
91 views

Ratio of 2 Gammas, approximation with power

Find all value of $\alpha$ such that $\lim\limits_{x\rightarrow +\infty}\left(\frac{\Gamma(x+\alpha)}{\Gamma(x)}-x^{\alpha}\right)=0$. (note: $\alpha$ is a constant with respect to $x$) By ...
3
votes
1answer
46 views

The number of holomorphic coverings (with given degree) of the punctured sphere is finite.

I'm looking for a proof of the following theorem: Fix a finite set $B=\{y_1,\ldots,y_k\}\subseteq \mathbb P^1(\mathbb C)$, then there is only a finite number of isomoprhism classes of ...
2
votes
5answers
497 views

$ABC=I\implies B$ is invertible and $B^{-1} = CA$

$A$, $B$ and $C$ are square matrices with $ABC=I$. I need to show that $B$ is invertible and $B^{-1} = CA$. I have proved it using the fact stated here. Since we only need to prove invertiblity of ...
0
votes
1answer
41 views

Different approaches for this problem (number in different bases)

I came across the following problem on a puzzle website the other day. A $3$ digit number is written $(xyz)_{10}$ in base $10$ and $(zyx)_9$ in base $9$. What is it? The website (which I think ...
2
votes
3answers
97 views

$x^{1+\epsilon}$ is not uniformly continuous on $[0,\infty)$

There are two questions. First: is the proof underneath correct? Let $\epsilon>0$ and let $f(x)=x^{1+\epsilon}$. I aim to show that $f$ is not uniformly continuous on $[0,\infty)$. We will show ...
2
votes
2answers
75 views

Epic-monic factorisation in $\mathbf{Set}$.

I'm stuck on Exercise 5.2.1 of Goldblatt's "Topoi: A Categorial Analysis of Logic": Given a function $f:A\to B$, if $h\circ g: A\twoheadrightarrow C\rightarrowtail B$ and $h'\circ g': ...
4
votes
0answers
87 views

${n \choose r}=\frac {n!}{r!(n-r)!}$ without using the permutation approach.

I had an idea that would be to first prove Pascal's Rule, $${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r},$$ which can be proved combinatorically whether one particular element (among the $n$) ...
0
votes
2answers
87 views

Proof that doesn't exists a rational $s$ such that $s^2 = 6$

Well, I solved it, and I would like to know if there is anything that can be corrected or improved here. I think that the proof ended up too long, and with too many letters. Surely there is a better ...
2
votes
1answer
28 views

Is this alternate answer right?

This is really basic, but I don't want to get the fundamentals wrong. Show that if $A$ is closed in $Y$ and $Y$ is closed in $X$, then $A$ is closed in $X$. The solution uses subspaces (which I ...
2
votes
3answers
29 views

Help complete proof ('or' statement in conclusion): $a^2 \ge 7a \Rightarrow a\le0 \text { or } a\ge7$

Question Prove that if $a$ is a real number such that $a^2 \ge 7a$ then $a\le0 \text { or } a\ge7$ My Attempt We are given: $a \in \mathbb{R}$ $a^2 \ge 7a$ And need to prove: $a\le 0 \text{ ...
0
votes
1answer
49 views

Give alternative proof of given theorem (relates to inverses of functions) need to use (b,a)∈ $B\times A.$

I have no idea of how to do this. We need to create an alternative proof using some of the ideas on the bottom, but I'm lost. Any ideas on how to do this? I'm not sure how to even start the ...
6
votes
1answer
213 views

Area of ​​the intersection of two discs: Integral solution?

Here is the problem : We consider two circles that intersect in exactly two points. There $O_1$ the center of the first and $r_1$ its radius. There $O_2$ the center of the second and $r_2$ its ...
5
votes
7answers
258 views

Alternative Proof of Infinitely Many Primes? [duplicate]

I've seen Euclid's proof of infinitely many primes, what are other approaches to proving there are infinitely many primes?
1
vote
1answer
57 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} = ...
1
vote
2answers
101 views

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

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 ...
0
votes
1answer
44 views

Convergence of Alternating harmonic series (Direct!)

Once again, note No use of the alternating series test!
0
votes
1answer
49 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 ...
4
votes
2answers
119 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 ...
1
vote
1answer
53 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 ...
11
votes
1answer
139 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 ...
2
votes
1answer
194 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 ...
6
votes
4answers
458 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 ...
0
votes
1answer
61 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 ...
1
vote
3answers
331 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 ...
28
votes
3answers
1k 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 ...
2
votes
1answer
71 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
votes
1answer
461 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 ...
14
votes
2answers
957 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 ...
0
votes
0answers
40 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 ...
0
votes
2answers
33 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 $N \cap K$ ...
0
votes
1answer
68 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 ...
0
votes
0answers
68 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: ...
2
votes
1answer
69 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 ...
1
vote
0answers
116 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 ...
0
votes
5answers
196 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 ...
3
votes
0answers
42 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, ...
0
votes
1answer
123 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 ...
3
votes
1answer
140 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 ...
1
vote
2answers
48 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 ...
0
votes
1answer
57 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 ...
1
vote
2answers
40 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 ...
0
votes
0answers
64 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 $|y|\ge y$, ...
2
votes
1answer
135 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 ...
1
vote
0answers
57 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 ...
6
votes
0answers
104 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 ...
12
votes
3answers
2k 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) ...