Back again! I only have two more questions, I promise... help is much appreciate as I seem to have found myself stuck and pretty much turned in a blank worksheet to my professor. He says these types of problems will be on our final, and I have no clue where to start.
Suppose we have a sequence of differentiable functions f$_n$:[-1,1] $->$ $\Bbb R$. We can assume for each $f_n$ that |$f_n$(x)|$\le2+$$x^2$ and |f ' (x)| $\le$ (2+$x^2$)/(1+$f(x)^2$). Why does the sequence (f$_n$) have a uniformly convergent subsequence. Quote any/all relevant theorems and prove that all the hypothesis are verified.
Hint: Apply the Arzela-Ascoli Theorem.
By the MVT, there is a point $\xi$ between $x$ and $y$ such that for $x,y \in [-1,1]$
$$|f_n(x)-f_n(y)| = |f_n'(\xi)||x-y|\leqslant \frac{2+\xi^2}{1 + f^2(\xi)}|x-y|\leqslant 3|x-y|.$$