I'm reading through Baby Rudin and I'm slightly confused about Theorem 3.6 (a).
The theorem is as follows:
$\textbf{ If $\{ p_n\}$ is a sequence in a compact metric space $X$,}$$\textbf{ then some subsequence of $\{ p_n\}$ converges to a point of $X$}$.
The main point of confusion for me is I'm not seeing how this is different than Theorem 2.37, stated in the previous chapter, which reads as follows:
$\textbf{If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$. }$
My thinking on it is that any subsequence in a compact metric space, or a compact set (which I believe may be regarded as the same), is by definition an infinite subset of a compact metric space. Indeed, even in the proof of Theorem 3.6 (a), where Rudin considers the case that the range of $\{ p_n\}$ is infinite, he refers back to Theorem 2.37.
However, I feel as if I'm missing something because if I've learned anything, is that Rudin is not one to write anything superfluous, so if it was the same, it certainly wouldn't be restated.
But ya, why is it even necessary to consider the separate cases of the range of $\{ p_n\}$ being finite and infinite, and what exactly is the difference between these two statements?
Thank you in advance for any help.
Best Answer
Let's consider the case when $\{p_n\}$ is infinite, and let's call this set $E$. If this subset has a limit point in $K$, it means there is a point $y \in K$ and a sequence of points $p_{i_1}, p_{i_2},p_{i_3},\cdots$ converging to $y$. However, this sequence may not be a subsequence of $\{p_n\}$ because we cannot assume that the indices are increasing, i.e. $i_1 < i_2 < i_3 < \cdots$. This explains the difference: Theorem 2.37 says that you can find some sequence in $E$ that converges, but a sequence in $E$ is not the same thing as a subsequence of $\{p_n\}$ due to the order of the indices.