Suppose that $U\subset \mathbb{R}^n$ is open and $u\in C^1(U)\cap H^1_p(U),1\leq p < \infty.$ For $V\subset\subset U$ and $0<\lvert h\rvert <<1$, we define
$$D_i^h u(x) := \frac{u(x+he_i)-u(x)}{h},\quad i=1,...,n.$$
I now want to understand why
$$\lvert\lvert D_i^h u - D_i u\rvert\rvert_{L^p(V)} \to 0, h\to 0.$$
My Professor stated that since $D_i^h u$ is measurable and converges pointwise to $D_iu$, this is a direct consequence of the dominated converge theorem. However, it isn't clear to me why there exists a function $g\in L_p(V)$ which dominates $\lvert D_i^h u\rvert$.
Similar questions have been asked before, but noone provided such a dominating function.