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 $r$.
J.C. Ottem
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