If you already have a proof for some result, but want to ask for a different proof (using different methods).
7
votes
5answers
483 views
Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]
Possible Duplicate:
Summation of a factorial
This equation is given:
$$
1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1
$$
I've solved it using mathematical induction but ...
6
votes
1answer
171 views
Proof that a certain entire function is a polynomial
Let $n\in\mathbf{N}$ be fixed, and $f$ entire and $|f^{-1}(\left\lbrace w\right\rbrace)|\leq n$ for every $w\in\mathbf{C}$. Then $f$ is a polynomial of degree at most $n$. I try to prove this ...
1
vote
1answer
142 views
Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic.
I was thinking that the product of groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}$ is not cyclic, but $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/p\mathbb{Z}$ is cyclic if p is an odd prime. ...
6
votes
1answer
564 views
Direct aproach to the Closed Graph Theorem
In the context of Banach spaces, the Closed Graph Theorem
and the Open Mapping Theorem are equivalent.
It seems that usually one proves the Open Mapping Theorem using the
Baire Category Theorem, and ...
5
votes
3answers
221 views
Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?
In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one?
I can of course ...
3
votes
3answers
402 views
$p=4n+3$ never has a Decomposition into $2$ Squares, right?
Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue's Lemma.
Is it correct to say that, primes of the form $p=4n+3$, never have ...
5
votes
1answer
242 views
Excessive use of the Yoneda lemma
In a MathOverflow thread on "nuking mosquitos", Andrej Bauer offered the following proof:
If two elements in a poset have the same lower bounds then they are equal by Yoneda lemma.
I understand ...
3
votes
0answers
197 views
Equivalence of Brouwers fixed point theorem and Sperner's lemma
I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. All proof I've found just prove that there must be at least one completely colored n-simplex, not that there must be ...
1
vote
1answer
122 views
Equivalences to “D-finite = finite”
By a D-finite set, we mean a set admitting no injection from the natural numbers (or equivalently, a set not in bijection with any proper subset).
I have encountered a proof that the following are ...
29
votes
1answer
1k views
$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?
If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and ...
25
votes
3answers
314 views
Alternative proofs that $A_5$ is simple
What different ways are there to prove that the group $A_5$ is simple?
I've collected these so far:
By directly working with the cycles: page 483 of ...
6
votes
1answer
439 views
How to deduce open mapping theorem from closed graph theorem?
These two theorems are equivalent but I can not figure out how to deduce the open mapping from the closed graph. Can anyone give a hint or some reference?
5
votes
3answers
387 views
Is this proof that $\sqrt 2$ is irrational correct?
Suppose $\sqrt 2$ were rational. Then we would have integers $a$ and $b$ with $\sqrt 2 = \frac ab$ and $a$ and $b$ relatively prime.
Since $\gcd(a,b)=1$, we have $\gcd(a^2, b^2)=1$, and the fraction ...
4
votes
1answer
263 views
Possible mistake in Krause's “Localization Theory”
I'm a little confused about a proof in this note by H. Krause; see page 12, Prop. 3.5.1. Can you explain me how the $\beta_i$'s are determined? I can't catch how to choose them in order to coequalize ...
3
votes
0answers
179 views
What are various proofs good for?
There are plenty of questions around here, which are proven to be right or wrong in various ways.
I wonder, what one can learn from these differing ways of how to prove something, despite the fact ...
3
votes
1answer
265 views
proof of l'Hôpital's rule that minimizes special-casing
A simple form of l'Hôpital's rule looks like this: If $u$ and $v$ are functions with $u(0)=0$ and $v(0)=0$, the derivatives $\dot{v}(0)$ and $\dot{v}(0)$ are defined,
and the derivative $\dot{v}(0)\ne ...
10
votes
1answer
909 views
The Hexagonal Property of Pascal's Triangle
Any hexagon in Pascal's triangle, whose vertices are 6 binomial coefficients surrounding any entry, has the property that:
the product of non-adjacent vertices is constant.
the greatest common ...
5
votes
4answers
188 views
Prove the following identity
I am having some trouble proving following identity without use of induction, with which it is trivial.
$$\sum_{n=1}^{m}\frac{1}{n(n+1)(n+2)}=\frac{1}{4}-\frac{1}{2(m+1)(m+2)}$$
I did expand the ...
3
votes
3answers
135 views
If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$.
In this (btw, nice) answer to Twin primes of form $2^n+3$ and $2^n+5$, it was said that:
If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$?
I'm not familiar with these kind of calculations, so I'd like ...
2
votes
0answers
76 views
Proof for a summation-procedure using the matrix of Eulerian numbers?
I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
