Why is it the $ x^2 + y^2 -\cos(z) <5 $ is unbounded, but $x^2 + y^2 + z^2 -\cos(z) <5 $ isn't?

Question

I am having a real hard time with this question. I know something is bounded if $\exists r>0$ s.t. $S\subset\beta(r,0)$

So with a function like $x^2 + y^2 - \cos(z) <5$. Why is it unbounded? Compared to $x^2 + y^2 + z^2 - \cos(z) <5$ ?

Answer 1

Since $-1\le\cos z\le 1$, if you have $x^2+y^2+z^2-\cos z<5$, then also $$ x^2+y^2+z^2<6 $$ and so the set $\{(x,y,z):x^2+y^2+z^2-\cos z<5\}$ is contained in the sphere centered at the origin with radius $\sqrt{6}$.

Conversely, take $x=0$ and $y=0$; then for any $z$ you have $$ x^2+y^2-\cos z<5 $$ which obviously makes the set $\{(x,y,z):x^2+y^2-\cos z<5\}$ unbounded.