Sequence of differentiable,equicontinuous functions I got stuck the other day trying to tackle the following problem : 
Let $ \left \{ f_n\left.  \right \} \right. $  be a sequence of differentiable functions : $ f_n \quad :  [0,1] \to \mathbb{R} $  with $ ||f'_n||_{\infty} \leq 1 $ 
Suppose  $$ \lim_{n \to \infty} \int^{1}_{0} f_n(x)g(x)dx=0  \quad (2)$$
For all continuous functions $ g:[0,1] \to \mathbb{R}  $
Prove that $ f_n $ converges uniformly to $ 0 $ . 
My progress so far is near trivial. I concluded using equicontinuity that it is sufficient to show pointwise convergence to some function $ f $ and then use dominated convergence to  $ (2) $  and then after some use of the problem's  linearity and weierstrass approximation theorem  with polynomials  conclude that $$ f {\equiv } 0 $$  .
I also got a trivial bound on $$ ||f_{n}||_{\infty} \leq \frac{1}{2} $$ ( if i didn't make any mistakes in calculations )  using the fact that $$ (f_n(x)-x)' \leq 0 $$ and $$ (f_{n}(x)+x)' \geq 0 $$ 
I can also obtain a uniformly convergent subsequence to $ 0 $ using Arzela-Ascoli but that doesn't seem to help much.
I would prefer , a slight/mild  hint to point me to the right direction , rather than a complete solution . 
 A: I am not sure the bound $\Vert f_n\Vert_\infty\le \frac12$ is correct. The sequence of constant functions $f_n(x)=\frac4n$ satisfies all your hypotheses.
Take $g=1$ and use the mean value theorem to find $x_n\in [0,1]$ such that $f_n(x_n)=\int_0^1f_n(x)\,dx\to 0$.  Using this and the bounds on the derivatives, you can prove that the sequence is equibounded and equi-integrable and then use Ascoli-Arzela. Is this a sufficient hint? It is not clear on where you are stuck.
Edit
Start from a subsequence $f_{n_k}$. Using Ascoli-Arzela you can find a durther subsequence $f_{n_{k_j}}$ which converges uniformly to some function $f$. Now use Weierstrass approximation theorem with polynomials to show that $f=0$.
Hence, for every subsequence $f_{n_k}$ you can find a further subsequence $f_{n_{k_j}}$ which converges uniformly to $0$. This implies that the original sequence must converge uniformly to zero. 
I am using a general fact about sequences in metric spaces. See the lemma in the link sequences
A: This is a "manual" solution to prove the pointwise convergence. I think one can skip the first two steps by taking 
$g$ to be a hat function.


*

*Show that there is a constant $M$ such that $\lVert f_n \rVert_\infty \leq M$ for all $n$.

*Show that we have in fact
$$ \lim_{n \to \infty} \int^{1}_{0} f_n(x)g(x)dx=0$$
for any $g \in L^1([0,1])$. (use a sequence $g_k$ of continuous functions that converges to $g$ in $L^1([0,1])$).

*Let $\varepsilon >0$, $a\in (0,1)$ and $g=1_{(a-\varepsilon,a+\varepsilon)}$ for $\varepsilon$ small enough. Prove that $f_n(a) \leq 2\varepsilon$ for $n$ large enough.

