Are these sets bounded and compact or not?

Question

I am interested in whether the following 2 sets are open, closed, bounded, and compact.

1) $A = \bigl\{(x, y) \in \mathbb{R^2}\quad|\quad|x − y| > 4\bigr\}$

2) $B = \bigl\{(x, y, z) \in \mathbb{R^3}\quad|\quad x + y − z \le 1, x^2 + y^2 + z^2 \ge 5\bigr\}$.

For 1), I have found it to be open, not closed, and so not compact by Heine-Borel. I suspect it is unbounded, but I am not sure how to show this. For 2), I have found it to be closed and not open, so it may be compact. I suspect it is bounded from the first constraint, but I am not sure how to prove this. My experience with proving boundedness and compactness is mostly limited to open ball definitions.

Answer 1

Both sets are unbounded. For the first one consider the points $(n,n+5),n=1,2...$ and for the second consider $(0,0,n),n=3,4,...$.

Answer 2

Your first set is unbounded indeed. For instance, it contains all the points if the form $(n,0)$, with $n\in\mathbb N\setminus\{1,2,3,4\}$.

The second set is also unbounded. It contains all points of the form $(n+1,0,n)$ with $\in\mathbb N\setminus\{1\}$.