Let \(\left\{X_{n}ight\}\) be a sequence of independent and identically distributed random variables where the distribution function of \(X_{n}\) is

\[F_{n}(x)= \begin{cases}1-x^{-\theta} & \text { for } x \in(1, \infty) \\ 0 & \text { for } x \in(-\infty, 1]\end{cases}\]

Define a new sequence of random variables given by

\[Y_{n}=n^{-1 / \theta} \max \left\{X_{1}, \ldots, X_{n}ight\}\]

for all \(n \in \mathbb{N}\).

a. Prove that the distribution function of \(Y_{n}\) is \[G_{n}(y)= \begin{cases}{\left[1-\left(n x^{\theta}ight)^{-1}ight]^{n}} & x>1 \\ 0 & \text { for } x \leq 1\end{cases}\]

b. Consider the distribution function of a random variable \(Y\) given by \[G(y)= \begin{cases}\exp \left(-x^{-\theta}ight) & x>1 \\ 0 & \text { for } x \leq 1\end{cases}\]
Prove that \(G_{n} \leadsto G\) as \(n ightarrow \infty\).