In 1917 Hardy and Ramanujan proved that all but $o(x)$ integers $n \leq x$ have $\log\log n + O((\log\log n)^{1/2 + \epsilon})$ distinct prime factors. The proof was long and relied on establishing (by induction!) an precise bound for the number of integers with exactly $k$ distinct prime factors (with $k$ arbitrary, and possibly tending to infinity with $x$). A short "two-line" proof was found by Turan in 1934.
Hardy disliked Turan's proof, because as he claimed, it did not give proper insight. However as it turned it was Turan's method that was prone to generalization. Twenty years later his inequality became the more general Turan-Kubilius inequality. Curiously enough it was later realized by Elliott that taking the "dual" of Turan-Kubilius's inequality yields immediately the arithmetic large sieve inequality! :-)
