Direct proof for "If for every sequence $(x_n)$ such that $\lim x_n = a$ it's true that $\lim f(x_n) = f(a)$, then $f$ is continuous"

Question

I already proved it by contradiction in the following way:

Negating $\lim_{x \to a} f(x) = f(a)$ assures the existence of $\epsilon>0$ such that $\forall n \in \mathbb{N}$ we can find $x_n \in X$ such that $0<|x_n-a| < 1/n \to 0$, but also $|f(x_n) - f(a)| \ge \epsilon$, which implies $\lim x_n = a,$ without $\lim f(x_n) = f(a)$, contradiction.

I'm looking for a direct proof of this result.