Proving Pseudo Random Generator from other Pseudo Random Generator? How can I prove that if $G$ is a Pseudo Random Generator, then other generator is a PRG (while the range's size is different)? 

For example: for the PRG $G$ (from $n$ to $t(n)$ where $n &lt; t(n)$), the generator $G'$ (from $n$ to $n+1$), where $G'$ returns the first $n+1$ bits of $G$) is PRG too?

Is the fact that the ranges are different affect the indistinguishability between the PRGs?
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 $D$.