I need to prove $f(y) \geq f(x) + \langle \nabla f(x), y-x \rangle, \forall x,y \in C \subset \Bbb{R}^n$ where $C$ is a convex set, and $f:C \rightarrow \Bbb{R}$ is a convex function.
I saw someone explain this and he reaches at the following expression using definition of convex function:
$$ \lim_{\lambda \rightarrow 0} \frac{f(x + \lambda(y-x)) - f(x)}{\lambda}\leq f(y) - f(x) \hspace{0.5cm} where,\hspace{0.1cm} \lambda \in [0,1] $$
I dont understand the next step:
$$\lim_{\lambda \rightarrow 0} \frac{f(x + \lambda(y-x)) - f(x)}{\lambda} = \langle \nabla f(x), y-x \rangle$$
Why is this true? Is this a definition?