showing existence of uniform convergent subsequence of functions (Arzela-ascoli applying?)

Question

Let $\{f_n\}$ be a sequence of real-valued $C^1$ functions on [0,1] such that for all n, $|f'_n(x)|\le 1/\sqrt x$
($0<x\le 1$) and $\int_{0}^{1}f_n(x)dx=0$

prove that $\{f_n(x)\}$ has a subsequence which converges uniformly on [0,1]

I think that I should apply arzela-ascoli theorem here, but I don't know how to start. How can i show equicontinuity and uniform boundedness of these functions?

Answer 1

Equicontinuity at every point of $(0,1]$ is clear from the Mean Value Theorem. Hint for equicontinuity at $0$: For every $\epsilon>0$ there exists $\delta>0$ such that $$\int_0^\delta\frac{dx}{\sqrt x}<\epsilon.$$

Hint for pointwise boundedness: If $f$ is $C^1$ and $\int_0^1f=0$ then $$f(x)=f(x)-\int_0^1f(t)\,dt=\int_0^1(f(x)-f(t))\,dt,$$so$$|f(x)|\le\int_0^1\int_x^t|f'(s)|\,dsdt.$$