Continuous Mapping is Sequentially Continuous

Theorem
Let $X$ and $Y$ be topological spaces, let $x \in X$.

Let $f : X \to Y$ be a function that is continuous at $x$.

Then $f$ is sequentially continuous at $x$.

Corollary
If $f$ is continuous on $X$ then $f$ is sequentially continuous on $X$.

Proof
Let $(x_n)_{n \ge 1}$ be a sequence in $X$ converging to $x$.

Let $V$ be a neighborhood of $f(x)$ in $Y$.

We are required to show that there exists $N \in \N$ such that $f(x_n) \in V$ for all $n \ge N$.

By continuity of $f$, choose a neighborhood $U$ of $x$ in $X$ such that $f(U) \subseteq V$.

Since $(x_n)_{n \ge 1}$ converges, there exists $N \in \N$ such that $x_n \in U$ for all $n \ge N$.

Therefore we must have that $f(x_n) \in f(U) \subseteq V$ for all $n \geq N$, as required.

Proof of Corollary
This follows immediately from the definitions:
 * 1) A function is sequentially continuous everywhere in $X$ if and only if it is sequentially continuous at each point
 * 2) A function is continuous everywhere in $X$ if and only if it is continuous at each point