Let $(f_n)$ a sequence of class $C^1$ at $[0,1]$, that is, functions whose first derivative is a continuous function. Suppose for all $n\in\mathbb{N}$ and for all $0< x\leq 1$ $$|f'_n(x)|\leq \frac{1}{\sqrt{x}}\quad\text{and}\quad \int_{0}^{1} f_n(t) \,dt=0$$
Show that $(f_n)$ has a convergent subsequence in $C[0,1]$.
I know that because Bolzano Weierestrass if I find bound for $(f_n)$ then it must has has a convergent subsequence in $C[0,1]$.
According to Mean Value Theorem - MVT, we have for $0 < x \le y \le 1$ and $n \in \mathbb N$
$$\vert f_n(x) - f_n(y) \vert \le \int_x^y \frac{dt}{\sqrt t} \le 2 \vert \sqrt y - \sqrt x \vert \tag{1}.$$
As $x \mapsto \sqrt x$ is uniformly continuous on $[0,1]$, we get that $\{f_n\}$ is uniformly equicontinuous on $[0,1]$.
Moreover the inequalities $(1)$ imply by continuity of $f_n$ that $$\vert f_n(y) - f_n(0)\vert \le 2$$for $0 \le y \le 1$. Therefore for all $y \in [0,1]$ and $n \in \mathbb N$:
$$f_n(0)- 2 \le f_n(y) \le f_n(0) + 2.$$
The equalities $\int_{0}^{1} f_n(t) \,dt=0$ have for consequence with the above that the sequence $\{f_n(0)\}$ is bounded by $2$ and finally that the sequence of functions $\{f_n\}$ is uniformly bounded by $4$ (for the $\Vert \cdot \Vert_\infty$ norm).
We get the desired conclusion by applying the Arzelà–Ascoli theorem.