Application of Arzela-Ascoli theorem: showing equicontinuity and pointwise boundedness

Let $$\{f_n\} \subset C((0,1))$$ be a sequence of functions such that

$$\sup_{n \in \mathbb{N}}\{f_n(0) + f_n'(0)\} =:M<\infty$$

and there exists $$\alpha \in (1, 2)$$ such that $$\forall n \in \mathbb{N}$$ and $$x \in (0, 1)$$,

$$|f''_n(x)|\leq 1/(1-x)^\alpha$$

Show that there is a subsequence that converges to a continuous function uniformly on every compact subset.

So we must prove $$\{f_n\}$$ is equicontinuous and bounded.

My calculus is a bit rusty so I'm struggling with this.

Is pointwise boundedness really as simple as saying, for every $$x \in (0,1)$$:

$$|f_n(x)| \leq \iint 1/(1-x)^\alpha dx = \frac{1}{(\alpha -1 )(\alpha - 2)(1-x)^{\alpha -2}}$$

for all $$n \in \mathbb{N}$$, and the RHS is independent of n, and thus is a bound?

I think that works, but I'm not sure how to show equicontinuity. I would appreciate the help.

Soft supplemental question:

What is the general approach and tricks used to prove stuff like this? I have been struggling with another question of the same sort (for which I would also appreciate help btw) and I would like to know the general strategy to prove boundedness and equicontinuity given these kinds of conditions on the functions.

• You are missing absolute value signs in the first condition. The way you have stated $f_n(x)=-n$ provides a counter-example. Aug 27 '20 at 5:36

$$f_{n}(s) = f_{n}(0) + f'_{n}(0)s + \int_{0}^{s} \int_{0}^{x} f''_{n}(t) dt dx$$.
$$f_{n}(s) - f_{n}(w) = f'_{n}(0)[s - w] + \int_{w}^{s} \int_{0}^{x} f''_{n}(t)dtdx$$ and the equicontinuity should be able to drop out from this.