Realcompact subspaces

We show that every compact space of large enough size has a realcompact subspace of size κ, for κ ⩽ c . We also show that an uncountable realcompact space whose pseudocharacter is at most ω 1 , has a realcompact subspaces of size ω 1 , thus, by continuum hypothesis, every uncountable realcompact space has realcompact subspace of size ω 1 .