Application of Arzela-Ascoli Thrm where $(f'_n)$ is uniformly bounded

Question

We have:

• Let $(f_n)$ be a sequence of differentiable functions on the interval $[0, 2]$
• Assume that $(f_n)$ converges pointwise to a function $f$ on $[0, 2]$
• Assume there exists positive real $M$ st $\forall n,$ $ \sup_{x\in[0,2]}\{|f'_n(x)|\}=M$

Prove that there is a strictly increasing sequence $(n_k)$ such that $(f_{n_k})$ converges uniformly on $[0,2]$.

I know that the Arzela-Ascoli theorem has the conclusion I want but to use it I need

1. $(f_n)$ to be uniformly bounded
2. $(f_n)$ to be uniformly equicontinuous

How would you fulfill those two conditions?

So following suggestions I have:

For 1:

Here we WTS $\forall n \in \mathbb{N}, x \in [0,2] |f_n(x)| \leq Y$ for some Y.

Pick arbitrary $n \in \mathbb{N}$ and $x_0 \in [0,2]$ then since $f_n \to f$ pointwise, $(f_n(x_0))_{n=1}^{\infty}$ is bounded thus $\forall n' \in \mathbb{N} |f_{n'}(x_0)| \leq Y.$ Then since $n\in \mathbb{N}$ we see that $f_n(x_0) \leq Y$ as required.

Then for 2:

let $\epsilon >0,\delta = \epsilon/M$ then $\forall x,y \in [0,2]$ and $n \in \mathbb{N}$ where $|x-y| < \delta,$ apply MVT to see $|f_n(y) -f_n(x)| = |f'_n(c)(y-x)| \leq M|(y-x)| \leq \epsilon$ as required.

Also as a followup do the condition applied here mean limit $f$ of $f_n$ is continuous?

Answer 1

You can use the bound on the derivative to get a uniform Lipschitz constant, which is more than enough for uniform equicontinuity; this follows from the mean value theorem or (if you prefer overkill) the fundamental theorem of calculus.

For the uniform bound, recall that a sequence of numbers that converges is bounded, so use that to get a bound on, say, $f_n(0)$. Then apply the uniform Lipschitz constant to propagate that bound across $[0,2]$.