Corollary of the Kolmogorov Zero-One Law, proof

Question

Here is an application of the Kolmogorov Zero-One Law given in my textbook (a probability path by Resnick page 107-108). It states that the random variables $\limsup_nX_n$ and $\liminf_nX_n$ are constant with probability 1 due to the law mentioned before. What I do not understand here is that the law only states that these two events should have probability 0 or 1. Could anyone clarify this, please? Thank you!

Answer 1

For a fixed $\lambda$, define the event $A:=\{\limsup_{n\to \infty}X_n\leqslant \lambda\}$. This is a tail event, hence it has probability $0$ or $1$. If we define $Y:=\limsup_{n\to \infty}X_n$, we thus have that for each $\lambda$, $\mathbb P\{Y\leqslant \lambda\}\in \{0,1\}$. Defining $c:=\inf\{\lambda, \mathbb P\{Y\leqslant \lambda\}=1\}$, we get that $Y=c$ almost surely.

The reasoning is similar for the liminf.