Function convergence's equivalence to sequence convergence

Let $X$ be a subset of $\mathbb{R}$, let $f:X\to \mathbb{R}$ be a function, let $E$ be a subset of $X$, let $x_0$ be an adherent point of $E$, let $L$ be a real, and let $\epsilon >0$ be a real. Then the following two statements are logically equivalent:

• $f$ converges to $L$ at $x_0$ in $E$.
• [...].