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 any function can be approximated by smooth functions .

The questions I have about the proof are the following:

It says choose so small that satisfies (3) in attachment. Is the reason that both conditions of (3) are satisfied for small enough because of two basic results of mollifiers which says that for fixed i it follows that as ?

Secondly, why is only defined for and not for ? Do we not need to cover 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 as can be seen in attachment?

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