Show all the sequence terms and the limit is compact in R^n

Question

Let $(a_n)$ be a sequence of $\mathbb{R^n}$ convergent (to $a \in \mathbb{R^n}$). Show that K=$\{a_n|n \in N^\star \} \cup \{a\}$ (all the sequence terms and the limit) is compact in $\mathbb{R^n}$.
($N^\star=\{0,1,2,3...\}$)

Answer 1

Since you are in $\mathbb{R}^n$, it is enough to show that $K$ is a closed and bounded. As a matter of fact, $K$ is trivially closed since it contains all its limit points. To see that is bounded, the idea is to understand that for a certain $n_0 \in \mathbb{N}$, all of the $a_n$ with $n\geq n_0$ are sufficiently near to $a$. Can you try to prove, bearing this in mind, that $K$ is a bounded subset?