This is a question about forcing. I have seen the following fact mentioned in multiple places, but have not been able to find a proof: if a random real is added to a transitive model of ZFC, then in the generic extension the set of reals in the ground model becomes meager.

My guess is that one should be able to, in some natural way, directly construct from a random real a countable sequence of nowhere dense sets covering the ground model reals, but I am not sure.