I want to sketch here the proof that if is a sequence of finite nonempty sets, and then has size for any nonprincipal ultrafilter on

The argument I present is due to Frayne, Morel, Scott, *Reduced direct products*, Fundamenta Mathematica, **51** (1962), 195–228.

The topic of the size of ultraproducts is very delicate and some open questions remain. For ultraproducts of finite structures, this is continued in Keisler, *Ultraproducts of finite sets*, The Journal of Symbolic Logic, **32** (1967), 47–57, and finally in Shelah, *On the cardinality of ultraproduct of finite sets*, The Journal of Symbolic Logic, **35 (1)** (Mar., 1970), 83–84. Shelah shows that if an ultraproduct of finite sets is infinite, say of size then His argument is a very nice application of non-standard analysis. The case that interests us is easier.

Clearly,

so it suffices to show that