a_{n} is a sequence of positive reals. h = lim (a_{1} + a_{2} + ... + a_{n})/n and k = lim (1/a_{1} + 1/a_{2} + ... + 1/a_{n})/n exist. Show that h k ≥ 1.

**Solution**

Apply the arithmetic-geometric mean theorem to each sum. We get that (a_{1} + a_{2} + ... + a_{n})/n ≥ (a_{1}a_{2} ... a_{n})^{1/n}, (1/a_{1} + 1/a_{2} + ... + 1/a_{n})/n = 1/(a_{1}a_{2} ... a_{n})^{1/n}. Hence their product is at least 1. Hence the product of the limits also.

© John Scholes

jscholes@kalva.demon.co.uk

15 Feb 2002