(Proximal) subgradient inclusion property proof

(Proximal) subgradient inclusion property proof References I\'m having a bit of trouble proving what seems to be two fairly straightforward statements for a nonlinear optimisation class I\'m taking. We\'re studying properties of the proximal subgradient, $\\partial_p(f) = \\{ v\\ |\\ \\exists \\rho, \\delta\\: f(y) \\geq f(x) + \\langle v, y-x \\rangle - \\frac{\\rho}{2}||y-x||^2\\ \\forall y = B_\\delta(x)\\}$ The two properties I\'m attempting to prove are as follows: If $f \\in C^1(\\mathbb{R}^n)$, then $\\partial_p(f) \\subseteq \\{\\nabla f\\}$ If $f \\in C^2(\\mathbb{R}^n)$, then $\\partial_p(f) = \\{\\nabla f\\}$ I\'ve managed to make a good start, I think, for the $C^1$ case -- replace $f(y)$ with $f(x) + \\langle \\nabla f(x), y-x\\rangle + o(||y-x||^2)$, divide through by $||y-x||$ and take $lim_{y \\rightarrow x}$. This takes me to $\\langle \\nabla f(x) - v, \\hat{h}\\rangle \\geq 0$ (where $\\hat{h}$ is some unit-length vector), which gives me that $\\nabla f(x) = v$ -- however, this gives me equality, not inclusion! Would someone be able to tell me what I\'m doing wrong, or what piece I\'m missing? At this point I\'m mildly confused, which is preventing me from making a start on the $C^2$ case!