As the title states, I am trying to determine if the given set $\{(x,y) \in \mathbb{R}^2 | 1 \leq x^2 + y^2 \leq 2 \}$ is compact / convex.
Progress so far:
Convex property:
This question has been taken from an old exam paper, where I used a double derivative in a previous question to determine the convex property for a $f''(x) \geq 0 \ \ \forall x \in [a,b]$. Can I apply a similar approach here?
Compact property:
For this to be satisfied it needs to be closed and bounded. I understand bounded to mean that the set is finite, and closed to mean that there are clear boundaries of the set. based on this (not particularly mathematical) intuition, I believe the set to be compact.
It is compact, it is not convex.
Closed and bounded subsets of $\mathbb R^2$ are compact.
The formal definition of compact is that every open cover has a finite subcover. But that is abstract and can be difficult to work with.
Another property of compact sets is that they contain all of their limit points. Or, every convergent sequence in the set converges to a point in the set.
A set is convex if every line segment between every two points in the set is in the set.
It is not convex as the line from $(-1,0)$ to $(1,0)$ passes a region not in the set.