okay this is how i went about solving the first one.. I would really appreciate input on if my thought process is correct since I am learning how to write proofs.

since A and B are compact subsets of Rn, A and B each have open covers where is an open cover of A (where scriptA is some index set), and is an open cover of B (where scriptB is another index set). Then there are and such that and which comes from the definition of compactness.

Adding A and B creates a new set C. In order for C to be compact, it must have its own set of open covers where the union of a certain collection of them (much like the union of open covers that contain A and B above) where . That means:

then since

also since

which implies so c is compact.

Input please! Thanks again in advance.