Showing the non-pseudorandomness of a generator I'm sorry if this question is too novice. It comes from a practice midterm in a cryptography course which I'm taking, and my confusion here is really that I don't know the "tools" that one uses in practice for this sort of question. The notation (and class) is Katz's. 

I've been given a PRG $G(s)$; I'm trying to show explicitly that $s\mathbin\|G(s)$ is not a PRG. Intuitively, I'd expect both components to be random-seeming, so there's nothing obvious that sticks out to me to XOR or manipulate.
This is an attempt to answer my own question based on Squeamish's helpful comments. If terminology is unclear or misleading, please tell me!

Let $D(s) = \mathbb{I}\{s = x||G(x), x \in \{0,1\}^n\}$. Then, if our putative PRG is $G'$, $\Pr[D(G'(s)) = 1] = 1$ - that is, $G'(s)$ is always in the format that $D$ looks for. However, $\Pr[D(r) = 1]$ (where $r$ is uniform from $\{0,1\}^{2n}$) is $\frac{2^n}{2^{2n}}$, to reflect that $s||G(s)$ has a range of only $2^n$ - being completely determined by $s$ - while there are $2^{2n}$ possible values $r$.

Thus $|\Pr[D(G'(s)) = 1] - \Pr[D(r) = 1]| = 1 - \frac{2^n}{2^{2n}}$, a non-negligible function.