Suppose that X1, X2 ... are infinite sequence of independent exponential random variables with mean u. Let X(n) = max{X1,...,Xn}. Show that the sequence X(n) is not bounded in probability.
I know the cdf of Fx(n) = [(1 - exp(-x/u))]^n not sure how to prove that it is not bounded in probality
For any $x>0$ $$\mathsf P(Y_n > x)=1-\mathbb P(Y_n\leq x)=1-\left(\mathbb P(X_1\leq x)\right)^n=1-(1-e^{-x/u})^n\to 1$$ as $n\to\infty$ since $(1-e^{-x/u})^n\to 0$.
So $Y_n$ converges to infinity in probability. As well as $Y_n\to\infty$ a.s. since $Y_n$ is nondecreasing sequence. So it is unbounded in probability.