Let $(x_n)$ be a divergent sequence in a compact subset of $\mathbb R^n$. Prove that there are two subsequences of $(x_n)$ that are convergent to different limit points.
Some ideas that might be helpful:
Heine-Borel theorem states that a subset of $\mathbb R^n$ is compact if and only if it is closed and bounded.
Bolzano-Weierstrass Theorem, every bounded sequence contains a convergent subsequence
A number $c$ is a limit point of $(x_n)$ if there exists a subsequence of $(x_n)$ convergening to $c$