Heine-Borel theorem for $\mathbb{C}^2$

Question

Lang's Complex Analysis contains a proof of the Heine-Borel theorem for $\mathbb{C}^1$, discussed previously on StackExchange.

In the reals, we know that the Heine-Borel theorem holds for $\mathbb{R}^n$. My question is: can Lang's proof be trivially extended to $\mathbb{C}^2$ or $\mathbb{C}^n$ generally?

Answer 1

Well, $\Bbb{C}^n$ is homeomorphic to $\Bbb{R}^{2n}$. So, if you know Heine-Borel for $\Bbb{R}^{2n}$, then you know it for $\Bbb{C}^n$.

Answer 2

Yes, the proof works exactly the same. In the case $n=1$, you get two sequences, one for the real part and one for the imaginary part, and you take convergent subsequences one after the other.

In the general case, you will have $2n$ sequences (one for each real and imaginary parts of the $n$ components) so now you have to take a convergent subsequence of the real part of the first coordinate; then a subsequence of this one for the imaginary part; and now you keep taking subsequences for each of the real and imaginary parts of the rest of the coordinates.