Let $X$ be a random variable with the property that $P(X\leq t)$ is either $0$ or $1$. Prove that $X$ is a.s. constant.
Suppose $\{X_n\}$ is an i.i.d. of random variables and $\{y_n\}$ is a sequence of positive numbers that tends to $+\infty$ as $n\rightarrow \infty$. The sequence $\displaystyle Y_n=\frac{X_1+X_2+...+X_n}{y_n}$ converges a.s. Prove it converges to a constant a.s.
For 1, what I need is there is a constant $C$ such that $P(X=C)=1$. Intutively it seems correct, but I have hard time proving it rigourously. Any hint/help is appreciated.
For 2. what I have is there is some function $Y$ such that $P(\lim_{n\rightarrow \infty} Y_n(\omega)=Y(\omega))=1$, and what I need to prove is $Y(\omega)$ is independent of $\omega$. But stuck on that. Any help is appreciated.