The following appears as an exercise in Bourbaki, Commutative Algebra, Chapter I, Exercises §3, n°1 (and other books). Consider a family of $R$-modules $(M_i)_{i\in I}$ ($R$ and $I$ arbitrary). We wish to show that $\bigoplus_{i\in I}M_i$ is faithfully flat iff all the $M_i$ are flat and at least one of them is faithfully flat.
A direct sum of modules is flat iff their summands are flat. Thus the if part follows from the fact that tensor products commute with direct sums. The only if part is where I am stuck.
My attempt goes with reductio ad absurdum: suppose none of the $M_i$ are faithfully flat, so there are for every $i\in I$ a non-zero module $N_i$ such that $M_i\otimes_R N_i=0$. From this, I'd try and build a non-zero module $N$ st $N\otimes_R\bigoplus_{i\in I}M_i = 0$, but I'm lacking ideas for such a candidate $N$. Even for $I$ reduced to 2 elements isn't obvious to me.
Is such an $N$ easy to find, or is there another way of going about this? Any suggestion is welcome. Thank you.