Prove compact sublevel sets imply coercivity

Question

$f$ is convex and $dom(f):x \in \mathbb{R}^N$. Define sublevel sets of $f$ as

\begin{equation} \mathbf{S}(f,\beta)=\{x \in \mathbb{R}^N\ : f(x) \leq \beta \} \end{equation}

are compact.

I need to prove that if $f$ is a convex function, then having compact sublevel sets is same as being coercive: for every sequence $\{x_k\}\subset\mathbb{R}^N$ with $\|x_k\|_2 \rightarrow \infty$, we have $f(x_k) \rightarrow \infty$.

How can I use the definition of sub level sets to prove this. Any help will be much appreciated.

Answer 1

Hint : if $S(f,\beta)$ is compact, there exist a ball with radius $R_{\beta}$ such that for all $x$ outside $B(0, R_{\beta}), f(x) > \beta$