Find limits of a function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ given by formula $f(x,y,x)=x+y+z$ on set $M=\left\{ (x,y,z)\in\mathbb{R}^3:x^2+y^2\le z\le 1 \right\}$. Does $f$ reaches all its limits?
To answer the last question I need to know if $M$ is a closed and bounded set. If it is then $f$ reaches all its limits by Bolzano-Weierstrass theorem. But I'm not sure if it is closed (bounded it is I think it just simply follows from inequalities, but probably it isn't precise proof). Previously I had only sets like $\left\{(x;y)\in\mathbb{R}^2: 4x^2+y^2\le 25\right\}$ and a reason for closedness was rather simple. I always used to state given set as an inverse image by continuous function of closed set. But in situation with $M$ I can't find similar explanation.
Finding limits seems to be hard too. I always used partial derivatives to find extremes in the interior of set and parameterization to explore boundary of set. But here I'm even not sure what will be interior and what will be boundary of $M$. Can anybody help?