Suppose that $f_{n}$be a sequence of real-valued functions that are uniformly Lipschitz and uniformly bounded on $\left[0,1\right]$,there exists constants $K,M>0$ such that for all $n\geq 1$ one has $|f_{n}(x)-f_{n}(y)|\leq K|x-y|$ and $|f_{n}(z)|\le M$,$\forall x,y,z\in \left[0,1\right]$. Prove that there exists a subsequence of $f_{n}$ that is uniformly convergent on $\left[0,1\right]$. Provide a counterexample if uniform boundedness is dropped.
I do not have any idea but I know all the definitions in the question.