Set $X = \lbrace u\in\mathcal{C}^2 [0,\pi]: u(0)=u(\pi)=0\rbrace$ equipped with the norm $$\Vert u \Vert = \left(\int_0^\pi (u'(x))^2\ dx + \int_0^\pi u(x)^2\ dx\right)^{1/2}$$
I want to prove that $(X,\Vert \cdot \Vert)$ is not complete. Can anyone give me a hint or some ideas to find a Cauchy sequence in $X$ that is not convergent in $\Vert \cdot \Vert$-norm to any element of $X$?. I spent a lot of time thinking but I couldn't find one.
Thanks in advance.