# Sequence in first-countable compact topological space

## Homework Statement

In a first countable compact topological space, every sequence has a convergent subsequence.

N/A

## The Attempt at a Solution

I'm self-studying topology, so I'm mostly trying to make sure that my argument is rigorous. I understanding intuitively why this argument doesn't work for a non-compact topological space, but I don't know why in a rigorous sense. I also have yet to take analysis, so I'm not very confident with sequences. Finally, I don't make use of the fact that the space is first-countable, which makes me suspicious.

Let $$X$$ be a first countable compact topological space, and let $$\{ p_n \}$$ be a sequence in $$X$$. Since $$X$$ is compact, we will pick two open sets $$U_{1_a}$$ and $$U_{1_b}$$ such that $$U_{1_a} \cup U_{2_a} = X$$ and $$U_{1_a}\cap U_{1_b} \neq X$$. By the pigeonhole principle, an infinite number of points in the sequence $$\{ p_n \}$$ will lie within one of these sets. Without loss of generality, we can say that this set is $$U_{1_a}$$. Within $$U_{1_a}$$, we pick one point of $$\{ p_n \}$$ that lies within $$U_{1_a}$$ and call it $$q_1$$ to being constructing a subsequence.

Next, we cover $$U_{1_a}$$ with two open sets $$U_{2_a}$$ and $$U{2_b}$$ such that $$(U_{2_a} \cup U_{2_b}) \cap U_{1_a} = U_{1_a}$$ and $$U_{2_a} \cap U_{2_b} \neq U_{1_a}$$. By the pigeonhole principle, an infinite number of points of the sequence $$\{ p_n \}$$ must lie within at least one of $$U_{2_a}$$ and $$U_{2_b}$$. We will label this set with $$U_{2_a}$$ without loss of generality. We pick a point of $$\{p_n\}$$ that lies within $$U_{2_a}$$ and label it $$q_2$$, declaring it to be the second point of the subsequence.

We consider with this process inductively, to obtain a subsequence $$\{q_n\}$$ that converges to some $$q$$ that lies within $$U_{1_a} \cap U_{2_a} \cap \ldots$$. Q.E.D.

I do not believe that this argument is correct because I do not make use of the fact that the space is first-countable. But, I do not know how to make use of that fact without being given a point that the subsequence converges to! So, where am I going wrong? I also do not think that I am making proper use of the fact that the space is compact, but I could be wrong.