Cartesian product of projective system

Question

Let $(M_i,\mu_i^j,I)$ and $(N_i,\nu_i^j,I)$ be two projective system of $R$-module ($R$ a commutative ring)

How to prove that : $$\varprojlim_{i\in I}(M_i\times N_i)\cong \varprojlim_{i\in I}M_i\times \varprojlim_{i\in I}N_i$$

Answer 1

If you already know that a limits commute with each other, you could deduce this one very easy without getting your hands dirty: A product is nothing else but a projective limit over an index-set $\{1,2\}$, where neither $1 < 2$ nor $2<1$ holds with respect to the partial ordering.

Hence product and limits commute.