# Proving a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$.

I have absolutely no idea how this is supposed to work, so an answer would be greatly appreciated!

Edit: Definition of compact is given as: A set S is compact if and only if every open cover of S has a finite subcover.

• What is your definition of compactness? – user99914 Oct 20 '14 at 21:40
• Do you know that subset of a metric space is compact iff any infinite sequence has a convergent subsequence? – Timbuc Oct 20 '14 at 21:41

A subset $K$ of a metric space $X$ is compact if every open cover of $K$ has a finite subcover.
Let $\{U_i\}_{i\in I}$ be an open cover of $K$. Then $p_0\in U_{i_0}$, for some $i_0\in I$. But as $U_{i_0}$ is open, then a whole ball $B(p_0,\varepsilon)\subset U_{i_0}$. Next, as $p_n\to p_0$, there exists an $n_0$, such that $n\ge n_0$ implies that $\varrho(p_n,p_0)<\varepsilon$ or equivalently $$n\ge n_0 \quad\Longrightarrow\quad p_n\in B(p_0,\varepsilon)\subset U_{i_0}.$$ Now all but (at most) the first $n_0-1$ terms belong to $U_{i_0}$. Next choose $U_1,\ldots, U_{n_0-1}$, so that $p_j\in U_j$, for $j=1,\ldots,n_0-1$. Finally now $$U_1,\ldots,U_{n_0-1}\,\&\, U_{i_0},$$ is a finite subcover of $K$.
Hint: Fix any element of an open cover of $K$ that contains $p_0$. Finitely many remain out there.