MathDB

Let

a_1 , a_2 , a_3 ,\ldots

be a sequence of positive real numbers, define

s_n = \frac{a_1 +a_2 +\ldots+a_n }{n}

and

r_n = \frac{a_{1}^{-1} +a_{2}^{-1} +\ldots+a_{n}^{-1} }{n}.

Given that

\lim_{n\to \infty} s_n

and

\lim_{n\to \infty} r_n

exist, prove that the product of these limits is not less than

1.

Problem Statement