According to my notes $V$ is coercive iff $\lim_{\|x\| \to \infty} V(x) = +\infty$. I have been translating this as $\forall M \in \mathbb{R}. \exists R > 0.\|x\| > R \implies f(x) > M$.
But then I don't see what norms should I be taking to prove:
If $V(x_1,x_2) = V_1(x_1)+V(x_2)$ then $V$ is coercive iff $V_1,V_2$ are coercive.
It seems really easy but I'm getting to the point of trying to find a counterexample (even if I think the proposition is true).