Thank you for your replies. I have been trying to finish off the question.

I tried latex the question in the forum but couldn't write multi-line equations using

Code:

\begin{align*}
\end{align*}

For part 1, I meant that the second last step is true because the function is in and since all other satisfying

are bounded above by

, where and is smaller than

where and . Hence the supremum is

unchanged by limiting the set to having just .

Is this correct?

For part 2, thanks for your clue, I put down the following.

Since and are both continuous, if at

some point , then there exists such that

.

Then any such that will ensure .

For part 4, I am not sure if the question meant to be supremum norm or essential supremum norm. For essential supremum norm, I don't think such an example exists.

Thanks a lot for helping me get through this question.