Let $\left\{f_{n}\right\}$ be a sequence of real value functions of class $C^{1}$ in $[0,1]$ such that $$\left|f'_{n}(x)\right|\leq \frac{1}{\sqrt{x}},\quad 0<x\leq 1,$$ and $$\int_{0}^{1}f_{n}(x)dx=0.$$ Show that $\left\{f_{n}\right\}$ has a subsecuence which is uniform convergent in $[0,1]$.
Remark: I think that the following theorem would be usefull.
Theorem: Let $K$ compact and $f_{n}\in\mathcal{C}(K)$ (that is the set of bounded and contiuous functions in $K$), then if we have that $f_{n}$ is puntually bounded for all $n$, and $\left\{f_{n}\right\}$ are equicontinuous, then $\left\{f_{n}\right\}$ has a subsecuence which is uniform convergent in $K$.
Hints: For equicontinuity use $f_n(y)-f_n(x) = \int_x^y f_n'(t)\,dt$ and the given estimate on the derivatives. For poinwise boundedness, note that each $f_n$ must be $0$ somewhere in $(0,1).$