- #1

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## Homework Statement

Suppose that ##a_n## is a sequence such that ##\sum_{n=1}^\infty a_n^2## converges. Show that ##\lim_{n\to\infty}a_n = 0##.

## Homework Equations

## The Attempt at a Solution

My idea was this. Since ##a_n^2## converges, we have that ##\lim_{n\to\infty}a_n^2 = 0##. I want to claim that ##\lim_{n\to\infty}a_n^2 = (\lim_{n\to\infty}a_n)^2 = 0##, but is this justified? Don't I have to know that ##a_n## does in fact converge before I can use the algebra of limits?