> 
> Prove $C[a,b]$ with norm $\Vert f\Vert=\left(\int\limits\_{a}^b\vert
> > f(t)\vert^2\right)^{\frac{1}{2}}$ is not Banach space.
> 
> 
> 

Banach space is complete normed space. We say complete normed space if every Cauchy sequence is convergent.

I have checked $\Vert f\Vert$ is norm. Now, I want to proof $C[a,b]$ is not complete under norm $\Vert f\Vert$, i.e. there exist Cauchy sequence which not convergent.

Now I confuse to choose the Cauchy sequence is not convergent. Can anyone give hint to choose Cauchy sequence is not convergent?