I'm having difficulties in following construction used in proof.
If $u$ is continuous function on open set $\Omega \subset R^n$ and $ p \in R^n$ satisfies $\displaystyle \limsup_{y\to x} \frac{u(y)-u(x)-p \cdot (y-x)}{|y-x|} \leqslant 0 $ then show that there exists a positive $\delta$ and continuous, increasing function $\sigma $ defined on $[0,\delta]$ such that $\sigma(0)=0$ and $u(y) \leq u(x)+p \cdot (y-x) + \sigma(|y-x|) \cdot |y-x|\mbox{ if }|y-x|<\delta.$
I thought $p$ can be understood as a gradient of a tangent line which lies above $u$, but I can't proceed it to construct such $\sigma$..