Convergent subsequences of sequences whose difference tends to zero Suppose we have two sequences $p, q:\mathbb{N} \rightarrow K$, where $K$ is a compact set and $K \subset \mathbb{R}$, such that $|p_n - q_n|\rightarrow 0$. Since $K$ is compact, there are $p'_n, q'
_n$ convergent subsequences of $p, q$ such that their limit are in $K$. Is it true that $|p'_n-q'_n|\rightarrow 0$, or in other words, that $\lim_{n\to\infty}p'_n = \lim_{n\to\infty}q'_n$? 

Since $K$ is a compact set $K \subset \mathbb R$ that means that $K$ is closed and bounded. Now, since $|p_n - q_n| \to 0$ this essentialy means that $p_n$ can approach $q_n$ infinitely closed and since they are $p,q : \mathbb N \to K$, then we can imply that they are bounded. But by the Bolzano-Weierstrass Theorem, each bounded sequence has a convergent subsequence, thus it is true that $|p'_n - q'_n| \to 0$.
