Gumbel distribution

A very short note for the Gumbel distribution.

$X_i$ i.i.d and have distribution of $\mathcal{Exp}(1)$, then $$\max_{1 \leq i \leq n}X_i - \log{n} \xrightarrow{(d)} G$$
where G represents a Gumbel distribution.

The proof is easy. We just calculate the distribution of this random variable
$$
\mathbb{P}[\max_{1 \leq i \leq n}X_i - \log{n} < y] = \mathbb{P}[\max_{1 \leq i \leq n}X_i  < y + \log{n}] = (1 - e^{- (\log{n} + y)})^n \rightarrow e^{-e^{-y}}
$$
So we conclude the convergence in distribution.