Proving Pseudo Random Generator from other Pseudo Random Generator?
Well I think you can prove it by reduction:

If there exist an efficient predictor $P$ that given the first $n$ bits of $G'$ is able to predict the $n+1$ bit then we can construct a distinguisher $D$ for $G$ defined as an elementary wrapper of $P$, that can distinguish $G$ from a truly random generator.
Since by assumption $G$ is a (secure?) pseudo-random generator there is not such a $D$.