For convex function $f$, its subdifferential at $x \in R^n$ is defined as \begin{equation} \partial f(x) = \{m \mid f(y) \geq f(x) + \langle m,y-x\rangle, \forall y \in R^n\}. \end{equation} Then if $f = f_1 + f_2$, where $f_1$ and $f_2$ are both convex, we have \begin{equation} \partial f(x) = \partial f_1(x) + \partial f_2(x). \end{equation} This notion can be generalized to $\epsilon$-subdifferential which is defined as \begin{equation} \partial_{\epsilon}f(x) = \{m\mid f(y) \geq f(x) + \langle m,y-x\rangle - \epsilon, \forall y \in R^n\}, \end{equation} where $\epsilon > 0$.
I find a relationship between $\partial_{\epsilon}f(x)$, $\partial_{\epsilon}f_1(x)$ and $\partial_{\epsilon}f_2(x)$ in a paper but without proof, which is \begin{equation} \partial_{\epsilon}f(x) \subseteq \partial_{\epsilon}f_1(x) + \partial_{\epsilon}f_2(x) \end{equation} I don't know how to prove it. Any hints or references?