Use Arzela-Ascoli: The pointwise boundedness follows from the bound on the supremum norm, and the equicontinuity from the uniform Lipschitz condition.
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.