Convergence of running maximum of uniform random variables Let $X_1, X_2, ... X_n$ be an IID sequence of IID random variables that have a uniform distribution $(0,1)$.
Let Max$(n) =$ max$(X_k:1\le k \le n)$, where $n\in \mathbb N$.
How do I show that Max$(n)$ converges to $1$ almost surely?
 A: Trivially we observe that the maximum of the list can only increase as we add more variables to it (for any given instance of the list).
$$\newcommand{\Max}{\operatorname{Max}} \forall n, 0\leq \Max(n)\leq 1, \Max(n+1)\geq \Max(n)$$
However, as each $X_i$ is a random variable, so to is $\Max(n)$.  So we need to examine the behaviour of the probability distribution of this.
$\begin{align}\Pr(\Max(n) > m) & = 1-\Pr(\bigcap\limits_{i=1}^n (X_i\leq m)) & \forall m \in [0, 1)
\\ ~ & = 1-\prod_{i=1}^n \Pr(X_i\leq m) & \text{by independence}
\\ ~ & = 1- m^n & \text{since } \forall i\in\{1..n\}, X_i\sim\mathcal{U}(0,1)
\\ ~ & ~
\\ \therefore \lim_{n\to\infty}\Pr(Max(n)\geq m) & = 1 & \forall m\in [0, 1) 
\end{align}$
Thus by squeezing, the probability that the maximum is 1 approaches certainty as $n$ positively approaches infinitude.
A: Let $Y_n = \max \{ X_1,\dots,X_n \}$. Then $Y_{n+1} = \max \{ Y_n,X_{n+1} \}$.
Consider $E[Y_{n+1} | Y_n]$. Note that $Y_{n+1} - Y_n \geq 0$, so
$$0 \leq E[Y_{n+1} - Y_n | Y_n] = E[Y_{n+1} | Y_n] - Y_n$$
So $Y_n$ is a submartingale. It is also nonnegative, so it converges a.s. by Doob's martingale convergence theorem. This does not state what the limit actually is.
To prove that the limit is $1$, suppose it were $y<1$. Then for all $n$, $X_n \in \left [0,y \right ]$. But by continuity of measure, $P(\forall n \: X_n \in [0,y])=\lim_{n \to \infty} P(X_1 \in [0,y], \dots, X_n \in [0,y])=\lim_{n \to \infty} y^n=0$. So our first step said that the probability it doesn't converge is zero, and the second step said the probability it converges to something other than $1$ is zero.
Of course this method of proving the first step is overkill, because you can instead just use that the actual sample paths themselves are nondecreasing and bounded, so you can use monotone convergence of sequences of real numbers, and then use the same method to do the second step.
