Unconventional (but instructive) proofs of basic theorems of calculus

I always love to prove that:

If $\{a_n\}_{n\in\mathbb{N}}$ is a bounded real sequence, it has a converging subsequence.

with the Erdos-Szekeres', or Dilworth's, theorem:

(Erdos-Szekeres, finite version) Every sequence with $n^2+1$ terms admits a weakly monotonic subsequence with $n+1$ terms.

(Dilworth, infinite version) Every infinite POset contains an infinite chain or antichain.

To prove Erdos-Szekeres, we send $n^2+1$ people to a post office with $n$ employees, $n$ queues. When a person arrives, he takes place in the first queue such that he is taller than the last person in the queue. If at some point someone ($A$) is not able to take place, then the people in the last position of every queue and $A$ give a decreasing sequence. On the other hand, if everyone is able to take place, there is a queue with at least $n+1$ people in it, giving an increasing sequence.

So we can use Erdos-Szekeres' or Dilworth's theorem to extract a (weakly) monotonic and bounded subsequence from $\{a_n\}_{n\in\mathbb{N}}$. Such a subsequence is clearly converging to its $\sup$ or $\inf$, and we are done.

I think the classic example of this is the whole field of non-standard analysis. It took 300 years to make infinitesimals rigorous (finally realized in the 1960's), but once equipped with such a toolkit you can derive all the basic calculus results (and much more) in just a couple of lines of infinitesimal algebra.