Vtyjkyuk

Let $f_n:[0,1]rightarrow mathbbR$ be continuous functions such that:
$|f'_n(x)|leq frac1sqrt x$ and $int_0^1f_n(x)dx =0$ for every $nin N$. Prove that exists a subsequence of $(f_n)$ which converges uniformly.

I know that I must apply Arzelá-Ascoli theorem. The problem then comes to show that the sequence $(f_n)$ is uniformly bounded and that it is equicontinuous.

What I have noted/done so far:

Since $|f'_n(x)|leq frac1sqrt x$, if $x,yin [0,1],xneq y$, then

$$|f_n(y)-f_n(x)|=|int_x^yf'_n(t)dt |leqint_x^y|f'_n(t)|dtleqint_x^yfrac1sqrt tdtleqint_0^1frac1sqrt tdt = K$$

but I can't use this to solve the problem. Any help? Thanks in advance

Observe that, for all $ninmathbbN$ and every $x,yin [0,1]$,
$$
|f_n(y) - f_n(x)| leq left|
int_x^y frac1sqrts, ds
right|
= 2 left| sqrty - sqrtxright|
leq 2 sqrt,
$$
so that the sequence $(f_n)$ is equicontinuous.

On the other hand, the condition $int_0^1 f_n = 0$ implies that the sequence is equibounded. Namely, from the mean value theorem, for every $n$ there exists a point $x_nin [0,1]$ such that $f_n(x_n) = 0$, so that
$$
|f_n(x)| = |f_n(x) - f_n(x_n)| leq left| int_x_n^x frac1sqrts, dsright| leq 2,
qquad forall xin [0,1].
$$