Consider the next set.
I have been ask to prove that is compact.
I really don`t know how to do it. I have no idea where to start from. Please help.
Here's an outline of an easy proof in this case:
1. Ennumerate the rationals in [0,1], use a diagonal argument to prove that there is a subsequence (of your function sequence) that converges at every rational.
2. Prove that this limit function (with the rationals as domain) satisfies the same Lipschitz condition.
3. Prove that there is an extension of this function to [0,1] and, finally, that it satisfies the conditions.