Series proof $\sum_1^\infty|a_n|<\infty$ then show that $\sum_1^\infty{a_n^2}<\infty$ [duplicate]

Iam stuck with this proof. There seems to be no property to help.
If $\sum_1^\infty|a_n|<\infty$ then show that $\sum_1^\infty{a_n^2}<\infty$ and that the reverse isnt true.

Solution 1:

The first series converges, so $|a_n|\to0$ as $n\to\infty$. Because of this, there exists an $N$ such that for all $n>N$, $|a_n|<1$. If $|a_n|<1$, then $a_n^2<|a_n|$. Eventually the terms of the new series are less than the terms of the old series. Then by direct comparison, the new series converges.

As for a counterexample to the reverse statement, other answers have already brought up the harmonic and over-harmonic series.

Solution 2:

$\sum_1^\infty{a_n^2} \leq (\sum_1^\infty|a_n|)(\sum_1^\infty|a_n|) < \infty$

A example to show reverse untrue.

See $a_n = \frac {1}{n} $

Solution 3:

Hints: Does $a_n^2\le |a_n|$ eventually hold? What do you know about the harmonic series?