Union of finitely many closed sets (a) Show that the union of finitely many closed sets is closed.
(b) Give an example showing that the union of infinitely many closed sets may
fail to be closed.

(a)
Given that $ \bigcup_{k=1}^{n} C_k $ is closed if $ \left[ \bigcup_{k=1}^{n} C_k \right]^c $ is open.
,and $ \left[ \bigcup_{k=1}^{n} C_k \right]^c $ is open if  $\bigcap_{k=1}^{n} C_{k}^c$ is open.
As It was shown previously that $\bigcap_{k=1}^{n} C_{k}^c$ is open (I did this in a previous post Intersection of open sets?),  $ \bigcup_{k=1}^{n} C_k $ has to be closed.
(b) any hint for this part?
Regarding (a) Is my argumentation sufficient? is there a better alternative?
Much appreciated.
Your proof of (a) is o.k.

(b): $ \bigcup_{k=1}^{\infty}[\frac{1}{k},1]$
The union of finitely many closed sets is open.
The intersection of an arbitrary collection of closed sets is closed.