Questions tagged [slick-proof]

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11
votes
0answers
817 views

Is there a slick proof of the fundamental theorem of dimension theory?

The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of ...
21
votes
8answers
3k views

Mathematical Writing: Proof Outlines/Overview in a Paper

While my question topic is that of mathematical writing of papers, which is a broad subject, the particular question is specific. I am writing a paper, in which we have a section called "Outline of ...
20
votes
5answers
2k views

Is Cauchy induction used for proofs other than for AM–GM?

The proof by Cauchy induction of the arithmetic/geometric-mean inequality is well known. I am looking for a further theorem whose proof is much neater by this method than otherwise.
11
votes
2answers
2k views

Reference for a nice proof of “undetermined coefficients”

I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along ...
5
votes
3answers
7k views

Proof without words for surface area of a sphere [closed]

I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the ...
16
votes
5answers
1k views

Smoothness of $f(\sqrt x)$

I found that I need to use the following facts in a paper that I am writing. Let $f\in C^\infty(\mathbb R)$, then If $f(0)=0$, then $f(x)=x g(x)$ for some $g\in C^\infty(\mathbb R)$. If $f$ is even, ...
1
vote
1answer
265 views

Synthetic Proof for Ratio of Volumes of Concentric Spheres?

Let $B^n(r)$ be the $n$-ball of radius $r$. A standard (easy) problem for first year calculus students is the following. $(1)$ Show that $$ \lim_{n\to \infty} \frac{\text{Vol}(B^n(r))}{\text{Vol}...
26
votes
6answers
2k views

Easy proof of the fact that isotropic spaces are Euclidean

Let $X$ be a finite-dimensional Banach space whose isometry group acts transitively on the set of lines (or, equivalently, on the unit sphere: for every two unit-norm vectors $x,y\in X$ there exist a ...
2
votes
2answers
237 views

A better way to compute the mapping spaces of the category of spans in an enriched tensored category?

Let X be a tensored and cotensored V-category, where V is a fixed complete, cocomplete, closed symmetric monoidal category. Define $C:=Span(X)$ to be the category of spans in X (this is the functor ...
80
votes
6answers
13k views

What are the most elegant proofs that you have learned from MO?

One of the things that MO does best is provide clear, concise answers to specific mathematical questions. I have picked up ideas from areas of mathematics I normally wouldn't touch, simply because ...
13
votes
2answers
2k views

Noncombinatorial proofs of Ramsey's Theorem?

I know of 2(.5) proofs of Ramsey's theorem, which states (in its simplest form) that for all $k, l\in \mathbb{N}$ there exists an integer $R(k, l)$ with the following property: for any $n>R(k, l)$, ...
59
votes
11answers
8k views

Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
3
votes
1answer
430 views

Slick verification of the model category axioms for Spaces and SSets with the q-model structure?

We choose our category of spaces to be compactly generated weak Hausdorff spaces for convenience, denoted $CGWH$. Questions: 1.) Is there any sort of slick argument to verify that CGWH with the ...
28
votes
2answers
1k views

Slick proof related to choosing points from an interval in order

Choose a point anywhere in the unit interval $[0, 1]$. Now choose a second point from the same interval so that there is one point in each half, $[0, \frac12]$ and $[\frac12, 1]$. Now choose a third ...
28
votes
6answers
7k views

A slick proof of the Bruhat Decomposition for GL_n(k)?

On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the ...
95
votes
3answers
20k views

Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional

A very important theorem in linear algebra that is rarely taught is: A vector space has the same dimension as its dual if and only if it is finite dimensional. I have seen a total of one proof of ...
329
votes
78answers
150k views

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...