prove that for a sequence $a_n$ from $n=1$ to $\infty$ that if the sum is less than infinity then $a_n$ tends to 0

Question

Prove that for a sequence $\displaystyle\sum\limits_{n=0}^\infty a_n$ that if $\displaystyle\sum\limits_{n=0}^\infty a_n < \infty$ then $a_n \rightarrow 0$.

Answer 1

Hints:

(1) We're given $\;\sum_{n=1}^\infty a_n=S\;$

(2) $\;a_n=S_n-S_{n-1}\;$

But (1), by definition, means $\;S_n\xrightarrow[n\to\infty]{}S\;$ , so ...