If $(f_n)$ is Cauchy in the $L^2$-norm, then is $(f'_n)$ Cauchy in the $L^2$-norm?

Question

Let $(f_n)$ be a sequence in $H^1(a,b)=\{f\in L^2(a,b);\;f'\in L^2(a,b)\}$, where $-\infty<a<b<+\infty$.

If $(f_n)$ is a Cauchy sequence in the norm $\|\cdot\|_{L^2}$, is it possible to conclude that $(f'_n)$ is a Cauchy sequence in the norm $\|\cdot\|_{L^2}$?

Thanks.

Answer 1

Consider $a=0$, $b=2\pi$ and $f_n(x):=c_n\sin(nx)$, where $(c_n)_{n\geqslant 1}$ is a sequence such that $c_n\to 0$ but $(nc_n)_{n\geqslant 1}$ is not bounded.