real analysis - Proving a set is compact!

This is the last one i need help with and the help is much appreciate as I seem to have found myself stuck and pretty much turned in a blank worksheet to my professor. He says these types of problems will be on our final, and I have no clue where to start.

Suppose I claim that the set {(x,y) $\in$ $\Bbb R^2$ : $e^x + e^y \le 100$ and x+y $\ge 0$} is compact. Prove this while also stating the theorems needed in obtaining the proof. I also think it would be helpful if anyone can verify why the hypotheses are satisfied if it isn't any bother...