Past open problems with sudden and easy-to-understand solutions What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved thanks to some out-of-the-box flash of genius, and the proof is actually short (say, one page or so) and uses elementary mathematics only?

The proof that if $f$ has absolutely convergent Fourier series and is never zero, then its inverse $\frac{1}{f}$ also has an absolutely convergent Fourier series.
Wiener gave a proof in 1932. Gelfand (1941) later developed the theory of Banach algebras to provide an elementary proof.
