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

# Questions tagged [alternative-proof]

3172 questions

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### product= $\exp\left[\frac{47\mathrm G}{30\pi}+\frac34\right]\left(\frac{11^{11}3^3}{13^{13}}\right)^{1/20}\sqrt{\frac{3}{7^{7/6}\pi}\sqrt{\frac2\pi}}$

$\mathrm G$ is Catalan's constant.
I recently found the product…

clathratus

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### A Geometric Proof of $\zeta(2)=\frac{\pi^2}6$? (and other integer inputs for the Zeta)

Is there a known geometric proof for this famous problem? $$\zeta(2)=\sum_{n=1}^\infty n^{-2}=\frac16\pi^2$$
Moreover we can consider possibilities of geometric proofs of the following identity for positive even inputs of the Zeta function:
$$…

finnlim

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### Characterization of positive definite matrix with principal minors

A symmetric 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 example, be found on…

Tara

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### What is the simplest proof of the pythagorean theorem you know?

Maybe enough so to explain it to children.

JEquihua

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### Proofs that every natural number is a sum of four squares.

I am planning to write a little note detailing several proofs of Lagrange's theorem that every natural number can be written as the sum of four perfect squares. I know of three different proofs so far:
a completely elementary proof by descent.
a…

Tony

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### Direct approach 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 then, from this theorem, proves the
Closed Graph…

André Caldas

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### Hahn-Banach theorem: 2 versions

I have a question regarding the Hahn-Banach Theorem. Let the analytical version be defined as:
Let $E$ be a vector space, $p: E \rightarrow \mathbb{R}$ be a sublinear function and $F$ be a subspace of E. Let $f: F\rightarrow \mathbb{R}$ be a linear…

KevinDL

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### Verify matrix identity $A^tD-C^tB=I$ on certain hypotheses

Given $n\times n$ real matrices $A,B,C,D$ such that:
$AB^T$ and $CD^T$ are symmetric
$AD^T-BC^T=I$
Prove that $A^TD-C^TB=I$
The solution I have come up with after a very long time is to consider:
$$\left( \begin{array}{cc}
A & -B \\
-C & D…

quangtu123

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### 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 integration. I've seen short and slick proofs of the…

Bachmaninoff

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### $\log_2 13$ is irrational

Is it true that $\log_2 13$ is irrational?
Let $x=\log_2 13\implies 2^x=13$.
So, it will be an irrational number, if not,$$x=\frac p q$$
and $$2^{\frac p q}=13$$
$$\implies 2^p=13^{q}$$
Since, $13$ is a prime number, $2^p$ divides $13^q$.
So, $2$…

David

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### "Novel" proofs of "old" calculus theorems

Every once in a while some mathematicians publish (mostly on the American Mathematical Monthly) a new proof of an old (nowadays considered "basic") result in analysis (calculus).
This article is an example.
I would like to collect a "big list" of…

Dal

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### Easy proof, that $\rm e\notin \mathbb Q$

$\def\e{{\rm e}}$
I recently had the task to explain the proof that $\e$ is irrational as a presentation to my classmates. To prepare this presentation, the teacher gave me a script with a proof that uses an estimation of the series $b_n =…

FUZxxl

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### 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 proof uses Axiom of Choice. My question is, if Axiom…

ILikeMath

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### Proof request: a collection of sliced squares of size 1 to n can always form a nontrivial rectangle

I'm an active member and challenge writer on Code Golf SE. Here is a challenge of mine, titled Make a rectangle from a collection of (sliced) squares:
Task
There is a famous formula on the sum of first $n$ squares:
$$ 1^2 + 2^2 + \dots + n^2 =…

Bubbler

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### Integral $\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$

Last year I wondered about this integral:$$\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$$
That is because it looks very similar to this integral
and this one. Surprisingly the result is quite nice and an approach can be found…

Zacky

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