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 ω1, has a realcompact subspaces of size ω1, thus, by continuum hypothesis, every uncountable realcompact space has realcompact subspace of size ω1.