At the moment I try to understand the topic "proving compact sets". 2 examples: I want to ask, if my assumptions/conclusions are right.
Example 1:
$(x_1-1)^3 + x_2 \le 0\:,\:x_2\ge0$
This set is closed because there are just greater/less than equal signs. This set is not bounded, because if $x_1$ goes to minus infinity, it is in the set.
Example 2:
$ x^2+y^2 \le 9\:,\: x^2 + y \ge 3$
This set is closed because there are just greater/less than equal signs. This set is also bounded, because negative values are not possible (because of the square is positiv, it is bounded by 9).
You are correct both times.
Of course you need to give an argument as to why they are compact, but drawing a picture often helps.
This is the first one in Wolfram Alpha, and this is the second one. Notice how the second one is just a disc, but where you have removed a parabola-shaped section of it.