I'm reading Complex Analysis by Serge Lang and I'm having trouble understanding the proof that he presented for the Heine-Borel theorem:
In particular, why is it that the presence of a subsequence converging to $a + ib$ implies that $S$ is compact? I can follow everything before that.
(The image is taken from this Math Stackexchange question, which unfortunately doesn't answer my question.)
Best Answer
In a metric space, you have the fact that sequentially compact is equivalent to compact, thus in $\mathbb{C}$ it suffices to show that any sequence in your set has a convergent subsequence, whose limit is in said set.