I am trying to prove from definition that a finite union of compact sets is compact given that the definition of an open cover I have from my lecture notes is:
An open cover $\cal U$ of a space $M$ is a collection of open subsets of $M$ s.t. their union is $M$.
P.S. I have proved the statement using sequential compactness but it's rather long. I've also seen many proofs of the fact but they all seem to use a different definition of an open cover.