Let $f,g:\mathbb{R}^n\rightarrow\mathbb{R}$ be two convex functions, continuously differentiable function with non-expansive gradients, i.e.
$$ \left\|\nabla f(x)-\nabla f(y) \right\|\le \left\|x-y \right\|\text{ for all }x,y\in\mathbb{R}^n$$
and
$$\left\|\nabla g(x)-\nabla g(y) \right\|\le \left\|x-y \right\|\text{ for all }x,y\in\mathbb{R}^n.$$
Does it mean that $\nabla f-\nabla g$ are also non-expansive? It is clear that without the convexity, this assertion will not correct by choosing $f(x)=-g(x)=-x^2$. However, I tried with examples for convex function and it is still true.