Show that $f(W_{t})$ is submartingale <=> $f$ is convex.

Question

Show that $f(W_{t})$ is submartingale ($W_t -$ Brownian motion) $\iff f$ is convex.

I can only show one way:

$\Leftarrow$

$f(W_{s})=f(\mathbf{E}(W_t|F_s))\leq \mathbf{E}(f(W_t)|F_s)$ and from definition $f(W_{t})$ is a submartingale.