the intersection of a collection of compact sets is compact

Hello

I have to prove that the intersection of a collection of compact sets is compact

This is what I have so far:

Each set in the collection is compact, thus each set is closed and bounded.

Each set is bounded if it is bounded above and below (i.e. there exists a B in R such that x <= B for every x in the set. There is an L in R such that x >= L for every x in the set.

Let Bj be the upper bound for each set j =1,...,n. Choose max (B1,B2,... Bj) =b. Thus, this b is the least upper bound of the collection of compact sets. Let Lj be the lower bound for each set j =1,...., n. Choose min (L1, L2,....Lj) =l. Then, l is the greatest lower bound of the collection of compact sets. Since the collection is bounded above and below, it is bounded.

Since each set in the collection is compact, each set is closed. Thus, the intersection of the collection of sets must be closed as well.

Since the intersection of the collection of compact sets is both closed and bounded, then the intersection is compact

what do you think?