Why if $\lim_{x \to a}f(x)=b$ then $\lim_{n \to \infty}f(x_n)=b$? Why is that if $$\lim_{x \to a}f(x)=b \Rightarrow \lim_{n \to \infty}f(x_n)=b?$$
This is often used to prove that the same properties that apply to sequences, also apply to functions, but I don't know from where this implication is coming from.
It is given that $\lim_{n \to \infty}x_n=a$ and that is derived from the fact that $a \in D'f$. I am not sure about this part either, t.i., why the $a$ being a limit point implies that there is a sequence within the domain that converges to $a$.

What the limit says is that however you approach the point $a$, the function $f(x)$ approaches $b$. Notice however you approach. That means in particular if you take any sequence $x_n$ which approaches $a$, then $f(x_n)$ must also approach $b$.
Formally if $f(x)\to b$ as $x\to a$, then by definition for all $\varepsilon>0$ there is a $\delta > 0$ so that $|x-a|<\delta$ implies $|f(x)-b|<\varepsilon$. Now suppose $x_n$ is a sequence which converges to $a$. Then for any $\delta>0$ there is an $N$ so that $n\geq N$ implies $|x_n-a|<\delta$. But since for these $n\geq N$ we have $|x_n-a|<\delta$ we must also have $|f(x_n)-b|<\varepsilon$. This 'proves' $f(x_n)\to b$ as $n\to\infty$.
It is however very important that $x_n\to a$, you cannot take any sequence (obviously).
