Prove that $S=((x,y) \in \mathbb{R}^2 : e^x+e^y\le 100$ and $x+y\geq 0)$ is compact I want to use the fact that a subset of $R^2$ is compact if it is exactly closed and bounded. For the closed part, I'm defining the continuous functions $f(x,y)=e^x+e^y$ and $g(x,y)=x+y$ and I'm saying 
$S=A\cap B$, where 
$A=((x,y)\in \mathbb{R}^2: 0\leq f(x,y) \leq 100$) and
$B=((x,y)\in \mathbb{R}^2: 0\leq g(x,y) \leq b$), with $b\in \mathbb{R}: B \supset S$ 
with these functions and the fact that the target set is closed and they are continuous,  then A and B are also closed and the intersection of closed sets is closed.
For the bounded part I don't know how to prove it. Any help with this guys? am I righ when proving S is closed?

You have that $e^x\leq e^x+e^y\leq 100$ so $x\leq\ln(100)$, and $y\leq\ln(100)$ too. As $x\geq -y\geq -\ln(100)$ and $y\geq -x\geq -\ln(100)$ you get $S\subset[-\ln(100),\ln(100)]^2$ which is bounded so is $S$.
