Proof of Sequential Criterion

Here is my attempt at a proof for the Sequential Criterion.

Definition: Sequential Criterion for Continuity

A function $f: A \to \Bbb R $ is continuous at the point $c \in A$ if and only if $\forall$ sequence $(x_n) \in A$ that converges to $c$, the sequence $(f(x_n))$ converges to $f(c)$.

Proof:

i) "$\rightarrow$"

Let $f: A \to \Bbb R$ be continuous at a point $c \in A$.Since $f$ is continuous at $c$ then $\forall$$\epsilon > 0$, $\exists$$\delta > 0 $ such that for $x \in A$ satisfying $|x-c| < \delta$$\implies$$|f(x) - f(c)| < \epsilon$.

ii) "$\leftarrow$"

Let $(x_n) \in A$ be a sequence converging to some point $c$.

If $(x_n) = $ constant function, then $f(x_n) = c$ and thus $f(x_n)$ converges to $f(c)$

If $(x_n)$ is $\neq c$, then an infinite subsequence $(x_n)_k, k\in \Bbb N$, can be formed in which the subsequence converges to $c$.

Therefore if $f$ is continuous at $c$, $f(x_n)_k) \to f(c)$, thus $f(x_n) \to f(c)$.

Any advice pointing out holes in my argument is appreciated!