2 Attachment(s)

Sobolev Spaces and some Mollifier stuff

Note that the proof in question is from book 'Partial Differential Equations by Lawrence Evans'.

Please view attachments for proof in question, basically I am trying to prove that for bounded $\displaystyle U$ any function $\displaystyle u \in W^{k,p}(U)$ can be approximated by smooth functions $\displaystyle u_{m} \in C^{\infty}\cap W^{k,p}(U)$.

The questions I have about the proof are the following:

It says choose $\displaystyle \epsilon_{i} > 0$ so small that $\displaystyle u^{i} := \eta_{\epsilon_{i}}\ast(\zeta_{i}u)$ satisfies (3) in attachment. Is the reason that both conditions of (3) are satisfied for small enough $\displaystyle \epsilon_{i}$ because of two basic results of mollifiers which says that for fixed i it follows that $\displaystyle u^{\epsilon_{i}} \rightarrow \zeta_{i}u$ as $\displaystyle \epsilon_{i} \rightarrow 0 \text{ a.e. } \text{ and if } \zeta_{i}u \in L^{P}(U) \text{ then } u^{\epsilon_{i}} \rightarrow \zeta_{i}u \text{ as }\epsilon_{i} \rightarrow 0 \text{ in } L^{P}(U) $?

Secondly, why is $\displaystyle W_{i}$ only defined for $\displaystyle i = 1, ...$ and not for $\displaystyle i =0$? Do we not need $\displaystyle \cup_{i=0}W$ to cover $\displaystyle U$ so that further in the proof together with a compactness argument we can show that there are finitely many non zero terms in the sum $\displaystyle v := \sum_{i=0}^{\infty}u^{i}$ as can be seen in attachment?

Thanks for any assistance, let me know if anything is unclear.