Let $F:\mathbb{R}^d \to \mathbb{R}$ be a convex function such that for all sequence $(x_n) \to \infty$ we have $||F(x_n)|| \to \infty$ $(*)$, where we use the euclidean norm here.
Level sets are sets of the form $\{x : F(x)\leq c\}$, where $c \in \mathbb{R}$. We know that the level sets are therefore also convex since $F$ is convex.
It is also clear from the property $(*)$ that level sets should be bounded. But how do we show that they are also closed? (and therefore compact)
If we know that convex functions are also continuous then we are able to prove this pretty easily, but can we do this without using that statement?