Points of increase for random walks

Say that a sequence S-0,...,S-n has a (global) point of increase at k if S-k is maximal among S-0,...,S-k and minimal among S-k,...S-n. We give an elementary proof that an n-step symmetric random walk on the line has a (global) point of increase with probability comparable to 1/log n. (No moment assumptions are needed.) This implies the classical fact, due to Dvoretzky, Erdos and Kakutani (1961), that Brownian motion has no points of increase.