Let $\displaystyle x_n$ a secuence of real numbers so that $\displaystyle \lim_{n \to \infty}{x_n} = x$. Prove that A = $\displaystyle ({x_n} : n \in \mathbb{N})$ $\displaystyle \cup$ $\displaystyle (x)$ is compact.
A non-empty set in R is compact if and only if it is closed and bounded (by Heine-Borel). Thus, it is sufficient to show $\displaystyle \{ x_n\} \cup \{ x\}$ is closed and bounded. It is definitely bounded because convergent sequences are bounded. The last thing to show it is closed which takes more work.
i think to show that A is closed we can use the argument that for a set to be closed it must contain all of its limit points (can be proven by contradiction, ie assuming it is not, then there's a subsequence converging outside the set, which gives us a contradiction). since our set is the values of the sequence x_n, and it contains the points it converges to (which is just one point x in this case as limit is unique in the topology of R). thus A contains all of its limit points (which is just x), and must be closed. since we showed bounded already, this set must be compact by heine borel theorem.
For $\displaystyle \sqrt{2}$ you gave a rational argument , but there is a problem with it. You are assuming that each sequence must go in order of the original sequence i.e. it is a subsequence. It does not have to.
Here is an argument that does not use Heine-Borel theorem, in fact, I think that doing this problem directly by definition is the way to go. Let $\displaystyle \{ S_i| i\in I\}$ be an open covering of $\displaystyle \{x_n\} \cup \{x\}$. Then $\displaystyle x\in S_{a}$ for some $\displaystyle a\in I$. Since this set is open there is an $\displaystyle \epsilon > 0$ such that if $\displaystyle |y-x|<\epsilon$ then $\displaystyle y\in S_a$ for all $\displaystyle y$ in $\displaystyle \mathbb{R}$. However, the sequence $\displaystyle \{x_n\}$ is convergent and so there is $\displaystyle N\in \mathbb{N}$ such that if $\displaystyle n>N$ then $\displaystyle |x_n-x|<\epsilon$ and by above it means $\displaystyle x_n\in S_a$. Now look at the points $\displaystyle x_1,x_2,...,x_N$. The point $\displaystyle x_1$ lies in some open set in $\displaystyle \{S_i\}$, say, $\displaystyle S_1$, and $\displaystyle x_2$ lies in say $\displaystyle S_2$, ..., and $\displaystyle x_N$ lies in say $\displaystyle S_N$. Then $\displaystyle \{ S_1,S_2,...,S_N,S_a\}$ is an open covering for $\displaystyle \{x_n\}\cup \{x\}$. We found a subcovering. Thus, $\displaystyle \{x_n\}\cup \{x\}$ is a compact set.