First proof(s) to ever make you think "nice"

well every irrational number can be the infinite sum of rational numbers.

ex. sqrt(2) = 1/1+4/10+1/100+4/1000+...

When I was in middle school I was able to prove there are infinite primes using math. I thought I was a genius but I was thousands of years too late.

Euclid's proof is much better.

I don't think that proofs are necessarily beautiful, rather what we can draw from those proofs.

I think that that the Sieve of Eratosthenes is the most stimulating, as it is simple in concept but has still stood the test of time for all these years. Even the Sieve of Atkin isn't as great.

Found the "artist" lampooned in OPs pic.

>So the harmonic series must diverge. It also nicely illustrates why the series diverges at logarithmic rate.
No, it doesn't. If it was an equality, it would

What was your proof? I never proved it, I just assumed it was true and took it for face value.

What is the point of this post? You don't actually think she would be on Veeky Forums.

Nice damage control.

The partial sums do grow logarithmically.

That may be the case, but it doesn't follow from that proof of it's divergence.

That's why I said it illustrates it, not proves.

Bolzano–Weierstrass theorem

proving the double angle identities in trig.

...

Model of supply and demand I remember was the first time I actually understood a part of how the universe works. Another one was proof of evolution. That shit was beautiful and pretty much explained all of the questions I had back then.

But even after my intense studies in physics and mathematics, I still cannot answer the first question I asked when I was about 5 years old: how do I get rich? Thats simply the most important solution a man should ever ask for.

true wealth comes from love of your friends and family :)

>true wealth comes from love of your friends and family :)
t. poorfag

Goursat integral lemma. It is a theorem that is Cauchy's integral theorem but for triangles. Our lecturer wanted to give us more intuition about Cauchy's theorem than by just proving it through Green's theorem, and it was very involved with geometry, and very interesting and long proof, that involves creating an infinite sequence of triangles that converge to a point.

i learned it that way too, was nice (from Stein/Shakarchi's book i think)

I really loved that class actually, because the lecturer was incredible, and really, all the results in complex analysis are surprising (liouville, holomorphic implies analytic, residue theorem), and it gives a lot of motivation for the study of algebraic topology/homotopy theory.

It may be because I havent seen that much algebra, but the deep theorems in analysis definitely have the best proofs

Nah, the people who say money doesn't matter are usually richfags who don't know what it's like to live without it.

But it is true. Money doesn't make you happy.

some studies show that money does make you happy, but suffers serious diminishing returns after 70k a year.

I've read that too, people making ~70k are frequently the happiest because they basically can get every fairly reasonable want they desire but aren't rich enough to be swamped in work or investments and aren't pestered/shamed/outed by poorfags

>proof of evolution

nice 1 dude; almost got me

eulers solution to the basel problem awed me the first time i saw it

The irrationality of sqrt(2) was also he first prove ever saw, but there is a far better version with prime decomposition, that also explains the n-th root and why it is either natural or irrational.

I actually have no idea what was the first I really liked, but I am reading euclids elements at the moment and it is full of really cool proofs.

money can give you freedom, which makes me pretty fucking happy :)

In calc 3 we proved that an open set is connected if and only if it's path connected. We separate the set into two disjoint open subsetst in a really neat way for a contradiction.

Euler's proof for the partition of a numbers using generating functions
Euler's derivation of the product over primes formula from the zeta function

Truly the master of us all

Sarah is a legit 10/10 the only reason she makes strips like the second panel is to leech off uggos

the implications you get from a set of numbers just by applying pigeonhole principle

the proof of abc conjecture

That there isn't any purely rational number p/q, for which p^2/q^2 = k. In other words proving for a square root of some integer to be irrational only requires you to find its decimal approximation, instead of doing a few more lines of equations about more number theory.

pythagoras theorem
sum of angles of a triangle
quadratic formula
sqrt(2)
definition of a limit

Infinity of primes
Uncountability of the reals

Godel's incompletude theorem

gynocentrism needs to be destroyed

nash's theorem

Here's a proof I didn't like, was that every ball of infinity norm is compact (proof by contradiction)

I'm not sure. It could have been that the inverse function of an increasing (or a decreasing) function is increasing (decreasing) proved using a chain [math]x

dumb animeposter

"if and only if" is not correct though, the Topologist's Sine Curve is a counterexample where it is connected but not path connected.

Polygonal path connected and we were in R^n

Plus it needs open I specified.

my mistake

Huh?

Proof that there exist irrational numbers x, y such that x^y is rational

pi ^ (1/pi) = 1

sigh...(You)

That is not an open set.

this is simply not true in all topological spaces. Maybe you did the case for R^n or some other simple pleb shit

>calc 3
>topological spaces
of course it was done in R^n you pretentious retard

Not really a "proof", but the first time I was genuinely astounded by math was in Calc II when my professor showed how Gabriel's Horn had a finite volume but a divergent surface area. Still hard for me to believe.