Finite union of compact sets is compact

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

Pf: Let $\displaystyle K = \cup_{i=1}^n K_i$ where $\displaystyle K_i $ is compact $\displaystyle \forall i

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Let $\displaystyle C$ be an open cover of $\displaystyle K$

$\displaystyle C$ is also an open cover for $\displaystyle K_1,K_2,...,K_n

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Since $\displaystyle K_i $ is compact $\displaystyle \forall i$, $\displaystyle \exists$ a finite subcover $\displaystyle C_i \: \forall i$

$\displaystyle \cup_{i=1}^n C_i$ is a finite subcover for $\displaystyle K$

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.