Function/R/Limit/Epsilon/Characterizations/Fact

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

$f:T⟶ℝ$

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

1. We have

   ${\lim }_{x\to a}f(x)=b.$

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

Write a proof