Let $f$ : $ \mathbb{R}^2 \rightarrow \mathbb{R}$ be a differentiable function. Prove that $f$ is Lipschitz continuous if and only if there exists $M>0$ such that $\|\nabla f(x,y)\|$ $\leq$ $M$ for all $(x,y)$ $\in$ $\mathbb{R}^2$.
I'm trying to prove Lipschitz continuity implies bounded gradient but I don't even know where to start. I know that there is a constant $L\geq 0$ such that
$ |f(x_1,y_1) - f(x_2,y_2)|$ $\leq$ $L$$\|(x_1,y_1),(x_2,y_2)\|$
But I don't know what to do next.