Lomonosov's 1973 proof that every compact operator $T$ has a hyperinvariant subspace (i.e., a subspace that is invariant for every operator that commutes with $T$) was much simpler than proofs existing then that every compact operator has an invariant subspace. See http://en.wikipedia.org/wiki/Invariant_subspace_problem. However, Wikipedia fails to mention that Lomonosov's proof was further simplified to replace the Schauder fixed point theorem by the spectral radius formula $\lim \|T^n\|^{1/n}$ (see e.g. Rudin's Functional Analysis), so that Lomonosov's theorem is taught (or assigned as an exercise) in classes in which the spectral radius formula is introduced.