Prove that a sequence of reals together with its limit is compact by cover I need to prove that the set $K =\{X_n: n\in\Bbb N\}\cup \{l\} $ is compact by covers. Where  $ l=\lim_{n \to \infty} X_n$. Where $X_n$ is a sequence of values in $\mathbb R$.

I understand that there are 2 cases. 
Case 1) K is countable and finite, where G = {$G_i$}$_{i\in I} $ is an arbitrary cove of K then $G^* = {G_i}_{n}$ is a finte subcover G. 

I need help analyzing the other case.
hint

Assume $$K\subset \cup_{i\in I} O_i $$

then
$$\exists j\in I \;:\; l\in O_j .$$

but $$l=\lim_{n\to +\infty} X_n $$

thus $$\exists N\in\mathbb N \;:\;n&gt; N \;\; \implies \;\;X_n\in O_j $$

and
$$K\subset (\cup_{p=0,1,2...N}O_p )\cup O_j $$

where $O_p $ is such that $$X_p\in O_p $$
hint

Assume $$K\subset \cup_{i\in I} O_i $$

then
$$\exists j\in I \;:\; l\in O_j .$$

but $$l=\lim_{n\to +\infty} X_n $$

thus $$\exists N\in\mathbb N \;:\;n&gt; N \;\; \implies \;\;X_n\in O_j $$

and
$$K\subset (\cup_{p=0,1,2...N}O_p )\cup O_j $$

where $O_p $ is such that $$X_p\in O_p $$