I need help in verifying/understanding a step in formulating an optimization problem used for support vector machines (though this question doesn't need any background in SVM). Consider a bunch of $m$ points $x_1, x_2,\ldots,x_m\in\mathbb{R}^n$, a vector $w\in\mathbb{R}^n$ and let $$\gamma_i=\frac{\langle x_i,w\rangle}{\|w\|} + \frac{b}{\|w\|}$$
where $b=-c\|w\|$ and $c$ is a constant. For this question, assume that $\gamma_i$ is positive for all $i$. Define $$\gamma\equiv \min_{i=1,\ldots,m} \gamma_i$$ and consider the optimization problem
$$\max_{w,b} \gamma \\\text{subject to}\ \frac{\langle x_i,w\rangle}{\|w\|} + \frac{b}{\|w\|}\geq\gamma $$
Letting $\hat \gamma=\gamma\|w\|$, the above can be rephrased as:
$$\max_{w,b} \frac{\hat\gamma}{\|w\|} \\\text{subject to}\ \langle x_i,w\rangle + b\geq\hat\gamma $$
Now we have $\hat\gamma=\min_i (\langle x_i,w\rangle + b)$ and assuming that $w'=w/\hat\gamma$ (which implies $b'=b/\hat\gamma$ since $w$ and $b$ are proportional), we see that $\min_i (\langle x_i,w'\rangle + b')=1$. So we can finally rephrase the optimization problem as (note that earlier we were maximizing w.r.t. $w, b$; now we're maximizing w.r.t. $w',b'$)
$$\max_{w',b'} \frac{1}{\|w'\|} \\\text{subject to}\ \langle x_i,w'\rangle + b'\geq 1 $$
Is this chain of reasoning correct or have I missed something/made an error in any step? Also, this problem is based on the following lecture. The optimization problem is stated on page 6 and the objective is $\max_{\gamma, w, b}\gamma$. Shouldn't it be $\max_{w,b}\gamma$, since we're only allowed to freely change $w$ and $b$?
The idea of this proof is essentially correct, the confusion about the difference between maximizing over $\gamma, w, b$ and over $w, b$ seems to be because there are two different possible ways to formulating the problem: One where you define $\gamma = \min_i \gamma_i$, as you do above. The other way is to specify constraints where $\gamma \le \gamma_i$ for all $i$ and optimize over $\gamma$ (which is essentially equivalent since the optimal choice of $\gamma$ will be the minimum of the $\gamma_i$).
The optimization problem as you have written it is sort of redundant, since $\frac{\langle x_i,w\rangle}{\|w\|} + \frac{b}{\|w\|} \ge \gamma$ is true just by your definition, and there is no need to write it out as a constraint. For it to be a linear program, technically you would have to do away with the definition that $\gamma = \min_i \gamma_i$, and just let $\gamma$ be a variable you optimize over.