How to use Poincaré inequality to prove $\|u\|_{L^2(I)}\leq C\|u'\|_{L^2(I)}$, $u\in H^1(I)$.

Question

I am studyng Sobolev spaces. In a class of functional analysis my teacher claimed the following:

If we consider an open interval $I=(l_0,l_1)\subset \mathbb{R}$ and define the Sobolev space

$$H_{l_0}^2:=\{u\in H^2(I):u(l_0)=u'(l_0)=0\}.$$

Then, using Generalized Poincaré inequality, we have that $\|u\|_{L^2(I)}\leq C\|u'\|_{L^2(I)}$.

We are using the book Direct Methods in the Theory of Elliptic Equations by J. Necas. This was pretty obvious to him, but it hasn't been to me. I know that $\|u\|_{L^2(I)}\leq \|u\|_{H^1(I)}$ and that since $u\in H^2(I)$ I have $u\in H^1(I)$. But I don't know how to proceed in order to prove the claim.

I asked this question before Here but in fact, I need the proof using Poincaré inequality.

Also, I have not been able to find what kind of Poincaré inequality the teacher used. Here I placed one that I found in a question asked on this platform, but I think that is not the one that was used by the teacher.

I know how to prove a similar claim but with $u\in H^1_0(I)$.

Can anyone help me with this problem?