**Question**

> 
> Let $f\_n:[0,1]\to\mathbb R$ be infinitely differentiable functions such that
> $$\lim\_{n\to\infty}\int\_0^1\!f\_n^2\,dx=0,$$
> and for all $\varepsilon>0$ there exists $N\in\mathbb N$ such that for all $n,m\geq N$,
> $$\int\_0^1\!(f'\_n-f'\_m)^2\,dx<\varepsilon.$$
> Must it be true that
> $$\lim\_{n\to\infty}\int\_0^1\!{f'\_n}^2\,dx=0?$$
> 
> 
> 

**Motivation**

The following is an exercise for a first course in functional analysis: Let $\Omega\in\mathbb R^n$ be compact. Prove that for all $f\in C(\Omega)$ there exists $u\in H^1(\Omega)$ such that
$$\int\_\Omega\!\nabla u\cdot\nabla\phi\,dx=\int\_\Omega\!f\phi\,dx\quad\text{for all }\phi\in C^\infty(\Omega),$$
where $H^1$ is the completion of $C^\infty$ under the inner product $\langle f,g\rangle=\int\_\Omega(fg+\nabla f\cdot\nabla g)\,dx$.

I can solve the question provided that $[f,g]=\int\_\Omega fg\,dx$ in $C^\infty$ extends to an inner product on $H^1$. If $(f\_n),(g\_n)$ are sequences in $C^\infty$ which are Cauchy (with respect to $\langle\cdot,\cdot\rangle$), it is not hard to show that $\lim\_{n\to\infty}[f\_n,g\_n]$ exists and is independent of choice of representative Cauchy sequences, so $[\cdot,\cdot]$ is well-defined in $H^1$.

Most of the axioms of an inner product space for $(H^1,[\cdot,\cdot])$ can be recovered from those in $(C^\infty,[\cdot,\cdot])$, except $[f,f]=0\implies f=0$. This is equivalent to the following statement about Cauchy sequences:

> 
> Let $f\_n\in C^\infty(\Omega)$ such that $[f\_n,f\_n]=\int\_\Omega\!f\_n^2\to0$, and $\forall\varepsilon>0\,\exists N\in\mathbb N\,\forall n,m\geq N$
> $$\langle f\_n-f\_m,f\_n-f\_m\rangle\to0,\quad\text{ie. }\int\_\Omega\!\nabla(f\_n-f\_m)\cdot\nabla(f\_n-f\_m)\,dx\to0.$$
> Show that $\langle f\_n,f\_n\rangle\to0$, ie. $\int\_\Omega\nabla f\_n\cdot\nabla f\_n\,dx\to0$.
> 
> 
> 

Taking the simplest case, $\Omega=[0,1]$, gives the question stated above.

**Edit**

> 
> I am looking for solutions with preferably minimal use of tools from functional analysis. In particular, the concept of weak convergence is only covered later in the course, and weak derivatives are not covered at all, so I suppose we are not expected to be familiar with them for this question.
> 
> 
> Right now it appears to me that the question is about a sequence of functions in $C^\infty$, or even $L^2$, so there should be a direct solution using tools from real analysis or measure theory. If this is incorrect, it would be very helpful if someone can answer the following (admittedly much more vague) question instead:
> 
> 
> **Question'**: why does the above question "live" in $H^1$, instead of $C^\infty$ or $L^2$?
> 
> 
>