Let a, b ∈ R with a < b, and let {$f_n$} be a sequence of differentiable functions from [a, b] to R. Suppose that both the sequences {$f_n$} and {$f'_n$} are uniformly bounded. Prove that the sequence {fn} is equicontinuous and has a uniformly convergent subsequence.
I am able to prove that {$f_n$} is equicontinuous using Mean Value theorem. Now for next part we can use the Azrela-Ascoli theorem with statement
If X is a compact metric space and F a subset of C(X), then F is compact if and only if F is closed, uniformly bounded, and equicontinuous.
Now [a,b] is compact and {$f_n$} is uniformly bounded and is equicontinuous also. Can somebody explain how can i prove that {$f_n$} is closed also ?
A trivial modification of the statment you have for Arzela-Ascoli theorem says the following:
If $X$ is a compact metric space and $F$ is a subset of $C(X)$ which is bounded and equi-continuous then it is relatively compact.
In a relatively compact set every sequence has a convergent subsequence.