Prove the equicontinuity of family of $C^1$ functions

Question

I'm trying to solve the following exercise:

Let $I=[0, 1] \subseteq \mathbb{R}$ and consider the family $$A= \{u \in C^1(I): ||u'||_{L^2} \leq 1 \}. $$ Prove that $A$ is a equicontinuous family.

Could somebody please give me a hint? Thanks in advance.

Answer 1

Hint: $u(x)=u(0)+\displaystyle\int\limits_0^xu'(t)dt$, $u(y)=u(0)+\displaystyle\int\limits_0^yu'(t)dt$, then $|u(x)-u(y)|=...$ (use Cauchy-Schwarz).