# Evans pde proof of global approximation theorem

I have a question regarding step 3 of the proof (shown in picture below).

I am wondering if there is a need for us to consider the subset $$V\Subset U$$ instead of directly using $$U$$ when showing the inequality in the last few sentence (i.e. using $$W^{k,p}(U)$$ instead of $$W^{k,p}(V)$$)

My idea is that we can consider $$v - u = \sum_i u^i - \sum_i \zeta_i u= \sum_i (u^i-\zeta_i u)$$ pointwisely a.e. because there is only finitely many nonzero term pointwise. Then we can continue the proof by considering $$\lVert v-u\rVert_{W^{k,p}(U)}$$ instead of $$W^{k,p}(V)$$.

Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem