Sequence of differentiable,equicontinuous functions

Question

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 .

Answer 1

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

Answer 2

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.