Let $f:X\to Y$ be an affine morphism of schemes. Prove that if a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent, then $f_*\mathcal{F}$ is also quasi-coherent.
I think the first natural idea is to take the exact sequence
$$\bigoplus_{i\in I}\mathcal{O}_X\big|_U\to\bigoplus_{j\in I}\mathcal{O}_X\big|_U\to\mathcal{F}\big|_U\to 0$$
and apply the direct image $f_*$ so that
$$\bigoplus_{i\in I}f_*\mathcal{O}_X\big|_U\to\bigoplus_{j\in I}f_*\mathcal{O}_X\big|_U\to f_*\mathcal{F}\big|_U\to 0$$
I've tried without success to prove that $f_*\left(\bigoplus_{i\in I}\mathcal{O}_X\big|_U\right)=\bigoplus_{i\in I}f_*\mathcal{O}_X\big|_U$. Maybe a finiteness condition (or something) is needed, but I can't see how $f$ being affine would guarantee that.
Any suggestions?