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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 $W^{k,p}(V)$.

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related : understanding proof of global approximation theorem Evans PDE

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

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