Let $(f_i)_{i\in\mathbb{N}}$ be a sequence of convex functions converging pointwise to another convex function $f:\mathbb{R}^n \to \mathbb{R}$. Suppose that at some point $x_0\in\mathbb{R}^n$, all the functions $(f_i)_{i\in\mathbb{N}},f$ have tangents. I define a tangent $l:\mathbb{R}^n\to\mathbb{R}$ for convex function $f$ at $x_0\in\mathbb{R}^n$ as an affine linear map satisfying $$ f(x) \geq l(x) := f(x_0) + c^T (x - x_0) $$ for some constant $c \in\mathbb{R}^n$ and for all $x\in\mathbb{R}^n$. Note that the tangent may not be unique at $x_0$ e.g. there is a kink in the function at $x_0$.
Question: Does there exist a sequence of tangents $(l_i)_{i\in\mathbb{N}}$ converging pointwise to $l$ where $l_i$ is a tangent for $f_i$ at $x_0$ and $l$ is a tangent for $f$ at $x_0$?
My attempt: We know that $f_i(x_0) \to f(x_0)$ from pointwise convergence but I dont know how to show how the gradient converges i.e. $c_i \to c$.