[Math] Suppose that the series $ \sum a_n$ converges. Prove that $\lim\limits_{n\to\infty} a_n = 0$.

Question

Suppose that the series $\sum a_n$ converges. Prove that $\lim\limits_{n\to\infty}
a_n = 0$.

Not sure how to do this my attempt:

WTS:$\forall \epsilon > 0, \exists N > 0$, such that for all $n \in \mathbb N$, if $n > N$, then $|a_n – 0| < \epsilon$

Let $\epsilon > 0$ be arbitrary

Choose N such that $|a_n – 0| < \epsilon$

Suppose $n > N$, then $|a_n – 0| < \epsilon$

I am not sure.

Answer 1

Suppose that $\sum\limits_{n=1}^{\infty}{a_n}=M$ then we have that $M=\lim_{n \to \infty}({a_n} + \sum\limits_{k=1}^{n-1}{a_k})=\lim_{n\to\infty}{a_n}+\lim_{n\to\infty}{\sum\limits_{k=1}^{n-1}{a_n}}=\lim_{n\to\infty}a_n+M$. Hence, we have that $a_n \to 0$.