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Write for simplicity $X=\mathbb{P}^n$. 2) follows since $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^r)$ is the global sections of the structure sheaf of the scheme $rY$ defined by $\mathcal{I}^r$ (which is connected). For 1), take cohomology of $$ 0 \to \mathcal{I}^r/\mathcal{I}^{r+1}\to\mathcal{O}_X/\mathcal{I}^{r+1} \to\mathcal{O}_X/\mathcal{I}^{r} \to 0. $$and note that the map $\Gamma(X,\mathcal{O}_X/\mathcal{I}^{r+1}) \to\Gamma(X,\mathcal{O}_X/\mathcal{I}^{r})$ is an isomorphism.