# Why is this series uniformly convergent on compact subsets

Given the series (family of functions) for every |z|<1 ($z\in \mathbb C$) we have, $$\sum_{n=0}^\infty z^n = 1 + z + z^2 + ... = \frac1{1-z}$$ This is given as an example of a 'normal family' of functions i.e. one who converges uniformly on compact subsets. I understand why it does not converge uniformly on $\mathbb B_1(0)$ but do not understand how it does on compact subsets (of $\mathbb B_1(0)$).

What would a compact subset of some set in $\mathbb C$ look like? [I never studied complex analysis :(] Then also why does it converge uniformly when we take said compact subset?

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I wlooks like the same as a compact subset of $\Bbb R^2$... –  Jean-Claude Arbaut May 19 at 11:20
For each $0 < r < 1$, the closed disk $K_r = \{ z : \lvert z\rvert \leqslant r\}$ is a compact subset of the unit disk. This family has the nice property that every compact subset of the unit disk is contained in some $K_r$. Since the series converges uniformly on $K_r$, it also converges uniformly on every subset of $K_r$. –  Daniel Fischer May 19 at 11:21
Ah, many thanks to you both. –  McT May 19 at 11:38