MathDB

8. Let

(a_n)_{n=1}^{\infty}

be a sequence of positive numbers and suppose that

\sum_{n=1}^{\infty} a_n^2

is divergent. Let further

0<\epsilon<\frac{1}{2}

. Show that there exists a sequence

(b_n)_{n=1}^{\infty}

of positive numbers such that

\sum_{n=1}^{\infty}b_n^2

is convergent and

\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}

for every positive integer

N

. (S. 8)

Problem Statement