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Write for simplicity $X=\mathbb{P}^n$.

An easy way of showing 1) is by noting that $\mathcal I / \mathcal I^2$ injects as a sub bundle of $\mathcal O_Y(-1)^{n+1}$ (this follows from combining the conormal sequence with the Euler sequence) and so none of it's symmetric powers $S^r(\mathcal I / \mathcal I^2)=\mathcal I^r / \mathcal I^{r+1}$ can have any global sections. Now taking the cohomology sequence of $$ 0 \to \mathcal{I}^r/\mathcal{I}^{r+1}\to\mathcal{O}_X/\mathcal{I}^{r+1} \to\mathcal{O}_X/\mathcal{I}^{r} \to 0, $$shows that $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r+1})$ injects into $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r})$, so by induction on $r$, we get 2).

Write for simplicity $X=\mathbb{P}^n$.

The easiest way of showing 1) is probably by noting that $\mathcal I / \mathcal I^2$ injects as a subbundle of $\mathcal O_Y(-1)^{n+1}$ (this follows from combining the conormal sequence with the Euler sequence) and so none of it's symmetric powers $S^r(\mathcal I / \mathcal I^2)=\mathcal I^r / \mathcal I^{r+1}$ can have any global sections. Now taking the cohomology sequence of $$ 0 \to \mathcal{I}^r/\mathcal{I}^{r+1}\to\mathcal{O}_X/\mathcal{I}^{r+1} \to\mathcal{O}_X/\mathcal{I}^{r} \to 0, $$shows that $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r+1})$ injects into $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r})$, and so by induction on $r$, we get 2).

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 $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r})$. Now 2) follows by induction on $r$.

Write for simplicity $X=\mathbb{P}^n$. 

An easy way of showing 1) is by noting that $\mathcal I / \mathcal I^2$ injects as a sub bundle of $\mathcal O_Y(-1)^{n+1}$ (this follows from combining the conormal sequence with the Euler sequence) and so none of it's symmetric powers $S^r(\mathcal I / \mathcal I^2)=\mathcal I^r / \mathcal I^{r+1}$ can have any global sections. Now taking the cohomology sequence of $$ 0 \to \mathcal{I}^r/\mathcal{I}^{r+1}\to\mathcal{O}_X/\mathcal{I}^{r+1} \to\mathcal{O}_X/\mathcal{I}^{r} \to 0, $$shows that $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r+1})$ injects into $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r})$, so by induction on $r$, we get 2).

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 $r$.

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 $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r})$. Now 2) follows by induction on $r$.