Given a PRF $F$ , is $G(s) = F_s(1)\|F_s(2)\|\cdots \|F_s(n+1)$ a PRG? If $F$ a PRF, and we construct $G$ using $F$ in the following way: 
$$G(s) = F_s(1)\|F_s(2)\|\cdots \|F_s(n+1)$$
where $|s|= n$. 

Is $G$ then a PRG? If so how can I prove this? If not how can $G(s)$ be distinguished from a random $r$? And what if $n+1 &lt;$ range of $F_s$? or it could be for any $n &gt; |s|$.
Consider the following game:



I assume certain familiarity with the definitions of Katz and Lindell book (this is exercise 3.14 in the book). In particular, $O$ is denoting an oracle. It can be either a random function oracle (if $b = 0$) or the output of a random function $F_s$ (if $b = 1$). What you do essentially in your distinguisher is query this oracle $l(n)$ times. This is a generalization with respect to your setting. Instead of $n$ one can write an arbitrary polynomial $l(n)$ since the involved algorithms are PPT.

In any case, note that when you gather all the $b_i$ you get $y = G(s)$ or $y$ uniformly distributed in $n \cdot l(n)$. This is because the independent concatenation of uniformly distributed bits is uniformly distributed (?). So, indeed what you have in the last part is a distinguisher for the PRG game. 

If you assume that $G$ is not a PRG then we can use an attack $A$ that has non-negligible probability $|P[WIN_{PRG}] - \frac 1 2|$. Then, we realize that one wins in the game PRF if and only if one wins in the PRG game (case distinction on $b$). Therefore, $|P[WIN_{PRF}] - \frac 1 2| = |P[WIN_{PRG}] - \frac 1 2|$ is non-negligible, but this contradicts the definition of PRF.