Can I use gradient operator on both sides of an equation?

Question

By first order approximation of Taylor series and finding the value of a function at x around y:

$f(x) \geq f(y) + \nabla f(y)^{T} (x-y)$

Can I do this:

$||\nabla f(x) - \nabla f(y)|| \geq || \nabla^2 f(y) ^{T} (x-y)||$

if $f$ is a convex function.