CPA-security of CTR mode I'm self-studying Katz' book (2nd ed), and I have a small question about Theorem 3.32, which says:


  If $F$ is a pseudorandom permutation, then $CTR$ mode is $CPA-$secure.


They present a proof there, but I think that another proof is viable, which is basically a corollary of a theorem they presented earlier. 

My proof is the following. Consider Theorem 3.31, which says


  If $F$ is a pseudorandom function, then $m\mapsto \langle r, F_k(r)\oplus m \rangle$ ($k$ is the key, $r$ is uniformly chosen) is a $CPA-$secure encryption scheme.


So, why don't we simply prove that $G_k(\cdot)$ defined as 
$$G_k({\tt ctr}) = \langle F_k({\tt ctr}+1), F_k({\tt ctr}+2),\ldots,F_k({\tt ctr}+\ell)\rangle $$
is pseudorandom? we would have by the cited theorem that $CTR$ mode (which is the encryption scheme described in the theorem using $G_k$) would be $CPA-$secure. 
Of course, showing that $G_k$ is pseudorandom if $F_k$ is does not impose a problem, since any distinguisher to $G_k$ can be turned into a distinguisher to $F_k$.

Everything makes sense, I don't see why this proof would not work and why did they bother writing a 2 pages long proof for what could be a corollary! (the only thing I think could fail is that the theorem used was proven for length-preserving $F_k$, but -I think- the proof also applies to the general case).

Anyone can point me to a mistake in my argument? btw, this could also prove that $OFB$ mode is $CPA-$secure.

Thanks in advance
One gap in your proposed argument is that $ctr$ is known to the adversary (it is included in the ciphertext), whereas saying "$G_k(ctr)$ is pseudorandom" implicitly assumes that both $k$ and $ctr$ are kept secret.  This is a minor point, because $G_k(ctr)$ is pseudorandom even if the adversary is also given $ctr$.

A more serious gap is that to prove CPA security, you must deal with the fact that the adversary can obtain multiple ciphertexts for messages of its choice.  If, for example, $ctr$ was always the all-zeros string (which would preserve the pseudorandomness of $G_k(ctr)$), the scheme would definitely not be CPA secure, because it would be breakable by a "two-time pad" attack.  So a correct proof must deal with the possibility that $ctr$ might be repeated from one message to the next, or more generally, that $ctr+i=ctr'+j$ for some two initial counter values $ctr, ctr'$ and offsets $i,j$ for two ciphertexts.  This issue is what most of the Katz-Lindell proof is dedicated to addressing, and is the source of the "birthday term" $q^2/2^n$ in the analysis.