# Questions tagged [slick-proof]

The slick-proof tag has no usage guidance.

17
questions

**11**

votes

**0**answers

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

**8**answers

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

**5**answers

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

**2**answers

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

**3**answers

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

**5**answers

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

**1**answer

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

**6**answers

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

**2**answers

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

**6**answers

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

**2**answers

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

**11**answers

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

**1**answer

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

**2**answers

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

**6**answers

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

**3**answers

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

**78**answers

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 ...