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...
Let $f(x,y)=e^x+e^y$, $f$ is continuous.
Let $g(x,y)=x+y$, $g$ is continous.
You set $A = \{(x,y)\in \mathbb{R}^2, f(x,y) \le 100, g(x,y) \ge 0 \} = f^{-1}(]-\infty;100]) \cap g^{-1}([0;+\infty[)$ is closed because $]-\infty;100]$ and $[0;+\infty[$ are closed, $f$ and $g$ are continuous and the intersection of two closed set is a closed set.
$e^y \le 100 - e^x \le 100$ so $y \le \ln(100)$ and, with the same method, $x\le \ln(100)$
$y \ge -x \ge -\ln(100)$ and $ x \ge -y \ge -\ln(100)$
And finally $-\ln(100) \le x \le \ln(100)$ and $-\ln(100) \le y \le \ln(100)$ so A is bounded
A is bounded and closed in finite dimension so A is a compact