Finite union of compact sets is compact

I think i have the proof, however I am worried about some assertions.

Pf: Let where is compact

Let be an open cover of

is also an open cover for

Since is compact , a finite subcover

is a finite subcover for

I am not sure if I can just say that an open cover of K exists. It makes sense, to me anyway, that I could cover any set with a bunch of open sets. Still, I am not sure if an open cover of K exists.

And the union of the finite subcovers remains finite?

Thanks for helping.