Function/R/Limit/Epsilon/Characterizations/Fact

Let ${}T\subseteq \mathbb {R}$ denote a subset and ${}a\in \mathbb {R}$ a point. Let

$f\colon T\longrightarrow \mathbb {R}$

be a function and ${}b\in \mathbb {R}$ a point. Then the following statements are equivalent.

1. We have

   ${}\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.$

2. For every sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ in ${}T$ which converges to ${}a$, also the image sequence ${}{\left(f(x_{n})\right)}_{n\in \mathbb {N} }$ converges to ${}b$.

Write a proof