TITLE: convergence of square summable sequence
QUESTION [3 upvotes]: I have seen a very interensting but for me confusing statement.
It says that if $\sum\limits_{n=1}^{\infty} a_{n}^{2} < \infty,$
then necessarily holds that $\lim\limits_{n \rightarrow \infty} a_{n}=0$ (from neccesary condition of convergence of sequence). But i don't understand, how is connected mentioned condition and convergence of squared summable sequence?  
I mean i know, that if the series of the sequence $a_{n}$ converges then $\lim\limits_{n \rightarrow \infty} a_{n}=0,$ but does it hold for the mentioned statement as well?

REPLY [1 votes]: You know
$$\sum_{n=1}^{\infty} b_n < \infty$$
$$\to \lim_{n \rightarrow \infty} b_n=0$$
Choose $b_n = a_{n}^{2} $. Then
$$\sum_{n=1}^{\infty} a_{n}^{2} < \infty$$
$$\to \lim_{n \rightarrow \infty} a_{n}^{2}=0$$
$$\to (\lim_{n \rightarrow \infty} a_{n})^{2}=0 \tag{You know why right?}$$
$$\to \lim_{n \rightarrow \infty} a_{n}=0$$
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