Finite Union of Compact Sets Clarification While studying some basic analysis/topology I have come across the proof regarding that the finite union of compact sets is compact using the definition of compactness. 
The proofs I have read all basically follow this:
For each compact set choose a finite subcover. The union of those subcovers will be finite and cover the union of the compact sets. 
Alright, not a big deal.
However, I think I may have a misconception regarding my notion of a compact set.
When proving a compact set is indeed compact must I not show that EVERY open cover has a finite subcover?
The aforementioned proof, to me, seams as though it is only considering one possible option for an open cover.
I am not doubting the validity of the proof, instead I am looking for some clarification as to why we do not consider the possibility of other open covers. 
 A: To clarify what the proof says: Suppose that you have compact sets $K_1, ..., K_N$ for some $N < \infty$. Choose any open cover $\mathcal{A}$ of $K_1 \cup ... \cup K_N$. Then by compactness, there is a finite subcover $\mathcal{A}_1$ of $K_1$; that is, $\mathcal{A}_1$ is a finite collection of open sets whose unions contains $K_1$. Choose a subcover $\mathcal{A}_2$ for $K_2$ in an identical manner, and continue to $\mathcal{A}_n$.
Then $\mathcal{A}_1 \cup ... \cup \mathcal{A}_N$ is a finite collection of open sets whose union covers $K_1 \cup ... \cup K_N$, so we've constructed a finite subcover.
Since $\mathcal{A}$ was arbitrary to begin with, we've covered all possibilites.
A: There are just a few details missing in the proof. Take any open cover of the finite union of compact sets. This cover is also a cover of each of the indiviuale compact sets. Now choose a finite cover for each of the compact sets. The finite union of these finite covers is a finite cover of the union of compact sets.
A: The proof should start as follows. Let $A_i$, $i=1$ to $n$ be compact. Let $\mathcal{C}$ be an open cover of $\bigcup A_i$. Then $\mathcal{C}$ is an open cover of any $A_i$. Let $\mathcal{C_i}$ be a finite subcover of $A_i$. Then $\dots$.  
