Discrete and compact subset must be finite Show that a discrete and compact subset $D \subset \mathbb{C}$ must be finite. Does this conclusion hold if $D$ is just discrete and bounded? How about discrete and closed?
Compact is the usual (for these simple spaces), closed and bounded, where closed is contained under the limit operation/contains all limit points.
Bounded it can be contained in a ball of some radius around the origin.
$D \subset \mathbb{C}$ is a discrete subset if $\forall z \in D$ there exists a ball of radius $r>0$ such that $D \cap B_r(z)$ = $\{z\}$.
Okay, for bounded set: why is discrete required? Can you give mme an example of a bounded set that is NOT finite?
 A: Proof without open covers:
Assume that $D$ is not finite, take an infinite sequence of distinct elements in $D$. The Bolzano-Weierstrass theorem (I hope you know this one) states that there is a subsequence , say $A_{n}$ that converges. But this is impossible, because a convergent sequence has a limit $L$ in $D$ (because $D$ is closed). But since $L\in D$ and $D$ is a discrete set, there is a $r \in \mathbb{R}$ so $B_{r}(L) \cap D = \{L\}$ This is in contradiction with the epsilon-delta definition of convergence. Take $\epsilon = r/2$ , then there is a $n \in \mathbb{N}$ such that norm of $(L-A_{i}) < r$ for each $i>n$ 
So D, can not be infinite.
A: If $D$ is discrete, then for every $z\in D$, there exists a $r_z>0$, such that $B(z,r_z)\cap D=\{z\}$, where $B(z,r_z)$ is the disc centered at $z$ with radius $r_z$.
Let now the collection of open sets
$$
{\mathcal C}=\{B(z,r_z): z\in D\}.
$$
Then $\mathcal  C$ is an open cover of $D$ without a proper subcover.
Since $D$ is compact, then the cover $\mathcal C$ is finite, and hence $D$ is finite.
A: Your question here contains a perfect example of a bounded and discrete set in $\Bbb R$ which is not finite. You should be able to extend that to $\Bbb C$. I think it should be a good exercise for you to identify which one it is. And this question also contains a proof for a compact, discrete set to be finite. But I'm afraid it does use open covers. But that is the only one I know. Apologies for that. 
And consider the set of all points in $\Bbb C$ which I believe you interpret to be the product set $\Bbb R \times \Bbb R$ which have integer coordinates. This set is closed, discrete but not finite. 
So in conclusion, NO.
- A set that is discrete and bounded is not necessarily finite.
 - A set that is discrete and closed is not necessarily finite.
