- #1

Zondrina

Homework Helper

- 2,065

- 136

## Homework Statement

If ##A## and ##B## are compact sets in a metric space ##(M, d)##, show that ##AUB## is compact.

## Homework Equations

A theorem and two corollaries :

##M## is compact ##⇔## every sequence in ##M## has a sub sequence that converges to a point in ##M##.

Let ##A## be a subset of a metric space ##M##. If ##A## is compact, then ##A## is closed in ##M##.

If ##M## is compact and ##A## is closed, then ##A## is compact.

Heine-Borel theorem : A subset ##K## of ##ℝ^n## is compact ##⇔## ##K## is closed and bounded.

I'm also told that a compact space is the best of all possible worlds :).

## The Attempt at a Solution

I'm told by the Heine-Borel theorem that a subset ##K## of ##ℝ^n## is compact ##⇔## ##K## is closed and bounded.

I'm thinking I should use this in particular to prove this because I'm told that ##A## and ##B## are compact. I believe this means I can bound the sets like so :

##min(A) ≤ A ≤ max(A)##

and

##min(B) ≤ B ≤ max(B)##.

The rest of this seems a bit too straightforward? Am I over thinking this or overlooking something?