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Let $f_n$: $[0,1]\to\mathbb{R}$ be differentiable satisfying both:

$$\begin{align} (1)& \int_0^1f_n(x)dx=0,\quad \forall n\in\mathbb{N},\\ (2)& |f'_n(x)|\le \frac{1}{\sqrt{x}},\quad\forall x\in(0,1]. \end{align}$$

Prove that $f_n$ has a uniformly convergent subsequence on $[0,1]$.

I want to apply Arzela-Ascoli Theorem, so I try to show that $f_n$ is uniformly bounded and equicontinuous. But I only get that $f_n$ is pointwise bounded. And I have no idea how to use the first condition.

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  • For boundedness: Fix $n$; there is $x_n$ such that $f_n(x_n)=0$ by the first condition. Then $$|f_n(x)|\leqslant\left|\int_x^{x_n}|f_n'(t)|dt\right|\leqslant\int_0^1t^{-1/2}dt.$$
  • For equi-continuity, the second condition gives that $|f_n(x)-f_n(y)|\leqslant 2\sqrt{|x-y|}$.

Note that since $f'_n$ is not assumed continuous, we can't apply directly fundamental theorem of analysis. But we can use the fact that $f_n$ is absolutely continuous, with a good condition on the derivative, to get what we want.

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    $\begingroup$ Ah, that makes much more sense. +1. $\endgroup$ – Alex Becker Jan 4 '13 at 22:25

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