Im reading Evans&Gariepy book measure theory and fine properties of functions second edition. I have a question about the proof of theorem 5.3 that I don't understand.
The theorem states that for a function $f \in BV(U)$ there are functions $f_k \in BV(U)\cap C^\infty(U)$ such that $f_k \to f$ in $L^1$ and $\|Df_k\|(U) \to \|Df\|(U)$. In the proof they definte the sets $$ U_k =\left\{x \in U: \text{dist}(x,\partial U) > \frac{1}{m+k}\right\} \cap B(0,m+k) $$ where $m,k$ are positive integers and define $$V_k = V_{k+1}\setminus V_{k-1} $$ and it is agreed that $U_0 = \emptyset$.
We can see that any element of $U$ is contained at most in 3 sets $V_k$, but What I don't understand is why this implies that $$\sum_{k=2}^\infty \|Df\|(V_k) \leq 3 \|Df\|(U-U_1) $$
I know that $\cup_{k=2}^\infty V_k = U\setminus U_1$ but I don't see why the last inequality is true. Any help is appreciated.