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 $a$.

If $(x_{n})_{n\in \mathbb{N}}$ is a sequence such that $\lim_{n \to \infty} x_{n} = a$
Then if $\lim_{x \to a} f(x) = b$ we have that for all $\varepsilon >0$ there exist $\delta >0$ such that if $|x-a|<\delta \implies |f(x)-b| < \varepsilon$
So, if we take a fixed $\varepsilon>0$, there exist a $\delta>0$ then by definition of limit, there exist $N\in \mathbb{N}$ such that if $n≥N$ then $|x_{n}-a|<\delta$ and that implies $|f(x_{n}) - b |<\varepsilon$
In other words, $\lim_{n\to \infty} f(x_{n}) = b$
