$C$ : a projective curve over an algebraically closed field, $I$ : the ideal sheaf of $\mathcal{O}_C$ defining a point $p\in C$. Why is the following hold?

$h^0(C,(\mathcal{O}_C/I^{r})\otimes\mathcal{O}_C(m))=dim(\mathcal{O}_{C,p}/m_{C,p}^r)\ge r$, where $\mathcal{O}_{C,p}$: the local ring at $p$ and $m_{C,p}$:the maximal ideal in $\mathcal{O}_{C,p}$.

the fact that $(\mathcal{O}_C/I^{r})\otimes\mathcal{O}_C(m)$ has support only at the point $p\in C$ implies "first equality". but I can't understand... and is "second inequality" from noetherian local ring?

I know my question is elementary..Please help me...