What is a good example of a proof whose technique is extreme overkill for the problem being solved?

Question

What are some proofs that are inappropriately powerful for the problem being solved? (Something like the math equivalent of using an atom bomb to kill a spider)

Answer 1

Problem: Prove that, for each $n\in\mathbb{N}\setminus\{1\}$, $\sqrt[n]{2}$ is irrational.

Proof: There is a very well-known proof for the case $n=2$. Suppose that $n>2$. If there were two natural numbers $p$ and $q$ such that $\sqrt[n]2=\frac pq$, then$$p^n=2q^n=q^n+q^n.$$This is impossible by Wiles' theorem (a.k.a. Fermat's last theorem).

Answer 2

Russell and Whitehead's proof that $1+1=2$ that took several hundreds of pages of lemma's in Principia Mathematica comes to mind.

I don't know if inappropriate is the right word in this case, since of course the very idea of P.M. was to show that you can build a lot of mathematics on top of very elementary axioms of logic and set theory, but it definitely feels like shooting a mice with a cannon.