I have a random sample of size $X_1, X_2, .., X_n$ following $U(0,2)$. I need to prove that $X_{(n)}$ which is the maximum ordered statistics will converge to $2$ in probability and almost surely.
I know that I can just prove this to converge to 2 almost surely and from there I can make a statement that since it converges almost surely, it will converge in probability and in distribution too.
But the definitions of convergence in probability and almost sure convergence looks identical to me. I could prove that this this maximum ordered statistic will converge to $2$ in probability using definition as follows:
$$\lim_{n \to \infty} P(|X_n-2|<\epsilon) = \lim_{n \to\infty} P(-\epsilon+2<X_n<\epsilon+2) = \lim_{n\to\infty} P(-\epsilon+2<X_n<2)$$
The CDF of the largest observation is $(\frac{x}{2})^n$. Hence, the above probability can be written as:
$$\lim_{n\to\infty} P(-\epsilon+2<X_n<\epsilon+2) = \lim_{n\to\infty}1 - F(2 - \epsilon) = 1$$
Hence, I could see that the largest observation will converge in probability.
How do I go about proving the almost sure convergence. I know the definition. I don't want to write it down again but I cannot separate between this definition and the one I have read for almost sure. It just puts the limit inside the probability statement unlike this definition which puts it before probability.