an is a sequence of positive reals. h = lim (a1 + a2 + ... + an)/n and k = lim (1/a1 + 1/a2 + ... + 1/an)/n exist. Show that h k ≥ 1.
Apply the arithmetic-geometric mean theorem to each sum. We get that (a1 + a2 + ... + an)/n ≥ (a1a2 ... an)1/n, (1/a1 + 1/a2 + ... + 1/an)/n = 1/(a1a2 ... an)1/n. Hence their product is at least 1. Hence the product of the limits also.
22nd Putnam 1961
© John Scholes
15 Feb 2002