When can one conclude that a sequence of uniformly bounded equicontinuous functions converges uniformly?

Question

Let $~f_{n}: [0,1] \rightarrow \mathbb{R}$ be a sequence of smooth functions that are uniformly bounded and equicontinuous. By Arzela Ascoli theorem we know that a subsequence $\{ f_{n_k} \} $ converges uniformly.

Is there is any additional condition under which one can say that the sequence $\{ f_{n} \} $ converges uniformly? In my case, I have sequence of functions that are uniformly bounded and the derivatives $f_{n}^{\prime}$ are also uniformly bounded. By fundamnetal theorem of calculus this implies the sequence is equicontinuous. Under what additional hypothesis can one conclude this sequence converges uniformly?

For example is this a sufficient criteria: $$ f_{n+1}(x) \geq f_{n}(x) \qquad \forall ~~x, ~~n $$ ?

Answer 1

A condition on the derivatives cannot guarantee the whole series (take $f_n(x):=(-1)^n$).

However, in the case where $(f_n(x),n\geqslant 1)$ is non-increasing for all $x$, and the sequence is uniformly bounded, this is true (and called Dini's theorem).

Answer 2

Here is one additional assumption that allows you to conclude: if there is a dense set $D\subset [0,1]$ such that the sequence $(f_n(z))$ is convergent for each point $z\in D$, then $(f_n)$ is uniformly convergent.

To prove this, first use equicontinuity to show that in fact the sequence $(f_n(x))$ is convergent for every $x\in [0,1]$ to some $f(x)$. This implies that the sequence $(f_n)$ has at most one limit point in $\mathcal C([0,1])$. Then apply Ascoli to conclude that $(f_n)$ converges uniformly to this only possible candidate.

For example, this works if the sequence $(f_n)$ is monotonic on a dense set of points (where the monotonicity is allowed to depend on the point). However, in this case the assumption of equicontinuity is not a priori needed (even though you do have equicontinuity a posteriori): it is not difficult to prove that monotonicity on a dense set of points implies monotonicity at each point, and then one can apply Dini's theorem separately to each of the two compact sets $K_1=\{ x;\; (f_n(x))\; \hbox{is nondecreasing}\}$ and $K_2=\{ x;\; (f_n(x))\; \hbox{is nonincreasing}\}$