Write for simplicity $X=\mathbb{P}^n$. An easy way of showing 1) is using inkspot's argument. Now 2) follows by the cohomology sequence of
$$
0 \to \mathcal{I}^r/\mathcal{I}^{r+1}\to\mathcal{O}_X/\mathcal{I}^{r+1} \to\mathcal{O}_X/\mathcal{I}^{r}  \to 0,
$$which shows that $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r+1})$ injects into $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r+1})$. Now 2) follows by induction on $r$.