When can one conclude that a sequence of uniformly bounded equicontinuous functions converges uniformly? 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: 
$$ f_{n+1}(x) \geq f_{n}(x) \qquad \forall ~~x, ~~n $$
? 
 A: 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).
A: 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 $K_2=\{ x;\; (f_n(x))\; \hbox{is nonincreasing}\}$
