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Sequence in first-countable compact topological space

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Homework Statement


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


Homework Equations


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

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

We consider with this process inductively, to obtain a subsequence [tex]\{q_n\}[/tex] that converges to some [tex]q[/tex] that lies within [tex]U_{1_a} \cap U_{2_a} \cap \ldots[/tex]. 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.
 

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