(2.7.3) If the series $\sum_{n=1}^{\infty} a_n$ converges then $(a_n) \rightarrow 0$.

For the absolute value function definition and other properties see here.
For the definitions of sequences and what it means to for a sequence to converge, see this, for subsequences see this.
For the definitions of series, and what it means to for a series to converge, see this.
For the “show the limit” template and an example, see this.

Formal Proof

Let $(a_n)$ be a sequence. We want to prove for any $\epsilon > 0$, there exists an $N \in \mathbf{N}$ such that

$$ \begin{align*} |a_n| < \epsilon \quad \text{whenever $n \geq N$}. \end{align*} $$

We are given that $\sum_{n=1}^{\infty} a_n$ converges. This means that for any $\epsilon > 0$, there exists an $N \in \mathbf{N}$ such that

$$ \begin{align*} |a_{m+1}+a_{m+2}+...+a_{n}| \quad \text{whenever $n, m \geq N$}. \end{align*} $$

Since this is true for any $n, m \geq N$, choose $n = m + 1$. Therefore,

$$ \begin{align*} |a_{n}| \quad \text{whenever $n, m \geq N$}. \end{align*} $$

From this we can conclude that $\lim (a_n) = 0$ as required. $\blacksquare$

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