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 P: 402 There are few things that are unclear in your post, but I think what you menat is this: (1) The set T is the topology of the space (X,p), that is, its elements are the open sets of (X,p). (2) In a general topological space, sequence convergence is defined by: $a_n \in X$ converges to $a \in X$ iff: $$\forall O \in T\exists n_0 \in \mathbb N \left(a \in O \wedge n > n_0 \rightarrow a_n \in O\right)$$ This means that, for any (open) neighborhood of $a$, all terms of $a_n$, for $n > n_0$ will belong to this neighborhood. (3) Now, you want to prove that, for any sequence, such that its set of terms $$$\left\{a_n:n \in \mathbb N\right\}$$ is infinite will have a convergent subsequence to p. For this topology, note that any neighborhood [itex]O_p$ of p will be a set such that its complement is finite; this implies that there exists an $n_0$, such that, for all $n> n_0$, $a_n \in O_p$. From this, you may extract a subsequence $a_{n_k}$, convergent, in the above sense, to p.