I am reading about extreme value theory. Let $X_1, X_2, ...$ be i.i.d. random variables and $M_n = \text{max}\{X_1,...,X_n\}$ as usual. I understand that, by the Fisher-Tippett-Gnedenko theorem: If there exist sequences $b_n \in \mathbb{R}$ and $a_n > 0$ s.t.
$$\frac{M_n - b_n}{a_n} \xrightarrow{d} G,$$
for some distribution function, then $G$ is a Generalised Extreme Value (GEV) distribution, i.e. belongs to the Gumbel, Frechet or Weibull family.
In a later chapter about the block maxima, there is the following statement in these notes extreme value theory (german, sorry!): "We model the distribution of the maxima through a GEV distribution. Of course, for the convergence normalising constants are needed, but in our case, we can assume that these are already part of the parameters $\mu, \sigma, \gamma$ (which are the location / scale / shape parameters of the GEV distribution."
This last sentence confused me. Does that hold in general? If yes, why? If not, which assumptions are neccessary?
Any hints appreciated. Thank you very much!
P.S. Like I said, I am referring to these notes on extreme value theory. It's German, sorry! (Remark 6.1.2) But the same statement can also be found here on these slides, for example (page 4).