Prove that the space of contraction mappings in $[0,1]$ is compact.

Question

I'm going through James Dugundji's Topology but I'm having trouble with chapter XI, section 4 problem 2:

2. Let $\mathscr{F}$ be the family of all continuous maps $I\to I$ such that $|f(s)-f(t)|\leq |s-t|$. Define $$d^+(f,g)=\max_{0\leq t \leq 1}|f(t)-g(t)|.$$ Prove: $\mathscr{F}$ is compact.

Here $I=[0,1]$. I'm familiar with the main properties of compactness and know that $1°$ countable paracompact spaces are compact iff every sequence has convergent subsequence. I was trying to use this on $\mathscr{F}$ but had no success.

I'm looking for any hints on how to start this proof.

Edit: I'm attempting an elementary proof, Arzela-Ascoli comes after this chapter so it shouldn't be used. Function spaces and completeness hasn't been touched on yet (I guess completeness in $I$ should be available though).