= Solution
Tensor the sequence from part (i) with the <twisting sheaf on projective space> $\mathcal O_X(n)$:
$$
0\longrightarrow\mathcal O_X(n-d)\xrightarrow{\cdot f}\mathcal O_X(n)
\longrightarrow i_*\mathcal O_Y(n)\longrightarrow0.
$$
Its <long exact sequence in cohomology> contains
$$
H^0(X,\mathcal O_X(n))\longrightarrow H^0(Y,\mathcal O_Y(n))
\longrightarrow H^1(X,\mathcal O_X(n-d)).
$$
Since $\dim Y\geq1$, we have $r\geq2$, and the final group is intermediate cohomology of projective space. It vanishes by the <cohomology of twisting sheaves on projective space>. Hence
$$
\boxed{H^0(X,\mathcal O_X(n))\twoheadrightarrow H^0(Y,\mathcal O_Y(n))\quad\text{for every }n\in\mathbb Z.}
$$
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