I was reading a proof in Evan's Measure Theory and he stated a lemma without proof that I cannot justify: Let $U$ be an open connected subset of $\mathbb{R}^n$
Let $U_k := \{x \in U\,|\, \text{dist}(x, \partial \Omega) > \frac{1}{k} \cap B_k(0)\}$ where $B_k(0)$ is the ball of radius $k$ centered at $0$. Then $$ ||Df||(U) := \sup \left\{\int_{U} f \mathrm{ div } \varphi\; |\; \varphi \in C_c^{\infty}(U), |\varphi| \leq 1\right \}\;\text{ implies }\;||Df||(U-U_k) \rightarrow 0. $$
It is easy to verify this for $f \in W^{1,1}(U)$ since we have $$ ||Df||(U) = \int_{U} |f'| dx. $$ But for the general case, I'm having problems verifying this lemma. Any help would be appreciated.