A confusion about a proof of that $f(x_n) \to f(x)$ then $x_n \to x$.

Question

In the book of Topology by Munkres, at page 130, it is given that

But I do not understand how can he concludes the converse result with the proof that he provided. I mean shouldn't we have chosen an arbitrary open set $U$ around $x$ and show that $\exists N $ s.t $\forall n \geq N$ $$x_n \in U \quad ?$$

Answer 1

No. The converse is this:

Let $f\colon X\longrightarrow Y$. If $X$ is metrizable and if for every convergent sequence $x_n\to x$, the sequence $f(x_n)$ converges to $f(x)$, then $f$ is continuous.

So, now he is assuming that $f$ maps convergent sequences into convergent sequences. And he wants to prove that $f$ is continuous, which he does using the fact that the continuity of $f$ is equivalent to the assertion that, if $A\subset X$, then $f\bigl(\overline A\bigr)\subset\overline{f(A)}$.

Answer 2

I assume there is a previous theorem that says something like

A function $f:X\to Y$ is continuous iff, for every $A\subseteq X$ we have $f(\overline A)\subseteq \overline{f(A)}$.