How can I prove/disprove that a construction yields a secure PRG? Let $G:\{0,1\}^{*} \mapsto \{0,1\}^{*}$ be a secure PRG. Prove or disprove that the following construction also yields a secure PRG.
$$G'(k) = G(k||0),$$
where $||$ denotes the concatenation of two strings.

I understand that proving/disproving such constructions usually involves a proof by reduction or a counterexample. Intuitively, I would say that $G'$ is a secure PRG, since we only fix a bit of the input, which should not make the output distinguishable (in polynomial time). However, for some reason I can't finde the right reduction, since I don't see how a $w \in \{0,1\}^{*}$ and a distinguisher $D'$ for $G'$ can be used to construct a distinguisher for $D$ for $G$.

So is $G'$ a secure PRG? If yes, how can I construct a valid reduction and if no, how does a distinguisher for $G'$ work?

I don't know if it is appropriate, but I would prefer hints, which point me in the right direction instead of complete solutions.
If there exists a secure PRG, then there is a secure PRG

whose last bit of output is always equal the last bit of its input.

If the last bit of G's output is always equal to the last bit of G's input, then the

function G' constructed as described in the question will not be a secure PRG.