Real Analysis: Function Continuity Let $A, B \subset \mathbb{R}$, let $f : A \rightarrow B$, and let $x_0 \in A$. Then $f$ is continuous at $x_0$ if and only if $\lim_{x \rightarrow x_0} f(x) = f(x_0)$.

I proved the first part 

$f$ is continuous at $x_o$ if $\lim_{x \rightarrow x_0} f(x) = f(x_0)$ 

using "$f$ is continuous at $x_o$ in $A$ ($ dom(f)$) if, for every sequence {${x_n}$} in  $A$ ($ dom(f)$) converging to $x_0$, we have $\lim_n f(x_n) = f(x_0)$"

Moreover, I tried proving the second part, which is 

$\lim_{x \rightarrow x_0} f(x) = f(x_0)$ if $f$ is continuous at $x_o$ 

using the other definition of continuity that involves $\delta &gt; 0$ and $\epsilon &gt;0$, which eventually proves that $\lim_n f(x_n) = f(x_0)$. 

My question is, is it correct proving the second part of the statement using aforementioned definition? or are there any other ways to prove the second part?
To prove the second part, suppose that it is not true. Then for some sequence $\{x_n\}$ converging to $x_0$, it is not true that $f(x)$ converges to $f(x_0)$, since otherwise for every sequence $\{x_n\}$ converging to $x_0$, $f(x)$ converges to $f(x_0)$, then $\lim _{x \to x_0} f(x)=f(x_0)$. Therefore $f$ is not continuous at $x_0$, which is a contradiction.
There is nothing to prove since it is just a definition that $f(x)$ is continuous at $x=x_0 \iff \lim_{x\to x_0} f(x)=f(x_0)$.

What we need to prove, in general, is that an assigned function $f(x)$ is continuous at a point according to the given definition.

Refer for example to the related


Continuous function proof by definition