Let $A$ be a commutative ring, and let $B,C$ be commutative $A$-algebras.
Let $M$ be a flat $B$-module and $N$ a flat $C$-module.
I want to show that $M\otimes_A N$ is a flat $B\otimes_A C$-module.
Here is my idea of the proof, I would like to know if it is correct:
Clearly, if $M$ and $N$ are finitely generated free then $M\otimes_A N$ is finitely generated free $B\otimes_A C$-module.
By Lazard Theorem, we may write $M = \varinjlim F_i$ and $N = \varinjlim G_i$ where $F_i$ and $G_i$ are finitely generated free.
Hence, since tensor product commutes with colimits,
$M \otimes_A N \cong \varinjlim F_i\otimes_A G_i$, so it is a direct limit of finitely generated free modules, so again by Lazard's theorem, it is a flat $B\otimes_A C$-module.
Is this reasoning correct? is there a simpler proof?