$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$

**Example**

Suppose that $a_n=c$ for all $n\in \N$. That is, $(a_n)$ is a constant sequence. Then, $\displaystyle \lim_{n\to \infty} a_n=c$.

**Proof**

Given any $\varepsilon$, let $N=1$. Then, for all $n>N$ we have

\[

|a_n-c|=|c-c|=0\lt \varepsilon .

\]