Prove that H(s) = G(!s) is a CPRNG if G(s) is one Given a cryptographic pseudo-random number generator $G$, prove that $H$ with $H(s) = G(!s)$ is also a cryptographic pseudo-random number generator. Here $s$ is a binary string; $!s$ is its complement.

For this we have to prove the two properties of a cryptographic pseudo-random number generator: the expansion and the pseudo-randomness.

The expansion is trivial to prove, because that's the same for $H$ and $G$. How can we prove that it is also cryptographically pseudo-random?
A proof by contradiction seems most appropriate here.

Assume that $H$ is not cryptographically secure. This means it fails to provide at least one of the properties of a CSPRNG as defined in your class, textbook, whatever. Then you must show how you can use an attack on $H$ to attack the cryptographic security of $G$. To do this, note that $H(!s)=G(!!s)=G(s)$. Since $G$ is cryptographically secure, your assumption must be incorrect. Therefore, you can conclude that $H$ is in fact cryptographically secure.