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I'm struggling with the following problem:

a) Let A be the set of complex numbers. A number z is called, as in the real case, a limit point, the limit point of the set A if for every $\epsilon>0$, there is a point a in A with $|z-a|<\epsilon$ but $z\neq a$. Prove the two-dimensional version of the Bolzano-Weierstrass Theorem: If A is an infinite subset of [a,b]$\times$[c,d], then A has a limit point in [a,b]$\times$[c,d]. Hint: use a bisection argument.

b) Prove that a continuous (complex-valued) function on [a,b]$\times$[c,d] is bounded on [a,b]$\times$[c,d].

c) Prove that if $f$ is a real-valued continuous on [a,b]$\times$[c,d], f is bounded on [a,b]$\times$[c,d].

What I did/know:

Not sure if my solutions for item a is formal enough.

a) As there are infinitely many points in A one can divide the closed rectangle in half, say "vertically", and, at least, one of the two halves will have infinitely many points in A. Taking this one with infinitely many points, divide it in half, "horizontally", one of the two halves has infinitely many points. Now, we have restricted an are with one-quarter of the size with infinitely many points. Doing this repeatedly, we can get a rectangle with infinitely many points with an arbitrarily small size- with diagonal less than $\epsilon$- such that taking the complex number at the center of the rectangle, call it $z$, every number in the rectangle is within $\epsilon>0$ of it.

b) Not sure what to do.

c) Not sure what to do.

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  • $\begingroup$ Do you know the real versions of these theorems, that any continuous real-valued function on a compact domain attains its maximum? $\endgroup$ – LutzL May 6 '18 at 13:33
  • $\begingroup$ @LutzL, I know them for $f:R \to R$, but I can't think of anything to generalize into that. $\endgroup$ – Fhoenix May 6 '18 at 13:39
  • $\begingroup$ Then do you know that $g(x)=\max_{y\in[c,d]}f(x,y)$ is again a continuous function? If not generalize the proof for the scalar case, assume it is unbounded and construct a sequence with $|f(x_n,y_n)|>n$ and consider a limit/accumulation point of it. $\endgroup$ – LutzL May 6 '18 at 16:06

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