Convex Optimization: Prove that a set of in-equalities is a bounded set

Question

I would appreciate any help or hints on how to proceed with a convex optimization problem. I have concluded that the set is constituted by following in-equalities, and want to show that this set is closed and bounded.

• $y^{2} ≤ 2xz$
• $x + yz + z^{3} ≤ 2$
• $z ≥ 0$

I don't see how the first equality puts an upper bound on y which I guess would be the best way to start approaching the problem.

Answer 1

Let $y=z=0$, $x$ can be as negative as possible. Hence the set is not bounded.