In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne.

To prove that the pushforward of a quasi-coherent sheaf is quasi-coherent, there are the following assumptions: either $X$ is noetherian, or $f$ is quasi-compact and separated.

Both these assumptions lead to some finiteness of affine covers on $X$ where we can write $\mathcal{F}$ as $\widetilde{M}$.

My question is: Why do we need these assumptions? In the proof, after reducing to $Y$ affine, the argument leads to the exact sequence

\begin{equation} 0 \rightarrow f_* \mathcal{F} \rightarrow \bigoplus_i f_* \left(\mathcal{F}|_{U_i}\right) \rightarrow \bigoplus_{i,j,k} f_* \left(\mathcal{F}|_{U_{i,j,k}}\right), \end{equation}

where the $U_i$'s are affines in $X$ and the $U_{i,j,k}$'s are affines covering the intersection $U_i \cap U_j$. The assumption (at least it's what I understood) are needed to make this family of indexes finite. But wouldn't the proof work even in the infinite case?

Of course I have to be mistaken somewhere, but I cannot see where and why!