Question

Prove that the convergence of
∑an (a sub-n)
an implies the convergence of

∑√(an)/n

if an > 0

Answer 1

Let\[\sum a_n\]be a convergent sequence. Then it is bounded. Moreover, by the Bolzano-Weierstrass theorem, it follows that since\[\sum_{n=1}^{\infty}\frac{\sqrt{a_n}}{n}\]is a subsequence of the above one, it converges as well.