Let $f$ be a $\rho$-lipschitz smooth function, meaning that $$ ||\nabla f(x) - \nabla f(y)|| \leq \rho ||x-y||, \forall x,y \in dom(f) $$
Is following inequality true?
$$ \nabla f(x) \leq \nabla f(y) + \rho ||x-y|| $$
How do I remove norm from the inequation and show that it is true?
It comes to me when I'm reading following math in an article
To be able to write inequalities with $\nabla f$ and for it to coexist and be summed with $\rho \|x - y\|$, it would need to have real values to begin with, in which case the left-side's norm is just the absolute value and it's fairly straightforward to go from the first inequality to the second.
The problem with that is that this can't apply to gradients for $f : \mathbb{R}^n \to \mathbb{R}$ with $n \geq 2$ since then the gradient of $f$ would be $\mathbb{R}^n$-valued and there's no "natural" meaning for $u \leq v$ when $u$ and $v$ are vectors. Yet that's not just a problem with what you wrote yourself but also with the text you quoted... Maybe the author used the $\nabla$ symbol for the divergence instead of the gradient?