Analysis with Programming has recently been accepted as a contributing blog on Mathblogging.org, a blogosphere aiming to be the best place to discover mathematical writing on the web. And as a first post, being a member of the said site, I will do proving on the theory of probability. This problem by the way, is part of my first homework on my masteral. This is my solution and if you find errors, do let me know.
Problem
- If $\{A_k\}$ is either expanding or contracting, we say that it is monotone, and for monotone sequence $\{A_k\}$, $\displaystyle\lim_{n\to \infty} A_n$ is defined as follows: \begin{equation}\nonumber \lim_{n\to \infty} A_n = \begin{cases} \displaystyle\bigcup_{k=1}^\infty A_k&\text{if}\;\{A_k\}\;\text{is expanding}\\[0.3cm] \displaystyle\bigcap_{k=1}^\infty A_k&\text{if}\;\{A_k\}\;\text{is contracting} \end{cases}. \end{equation} Prove the above equation.
Solution
- Proof. If $\{A_k\}$ is either expanding or contracting, then for an infinite sequence $A_1,A_2,\cdots$ one can define two events from $\displaystyle\lim_{n\to \infty}A_n$, i.e. \begin{equation} \label{eq:limAn} \lim_{n\to \infty} A_n = \begin{cases} \displaystyle\lim_{n\to\infty}\sup_{k\in [n,\infty)}\{A_k\}\\[0.3cm] \displaystyle\lim_{n\to\infty}\inf_{k\in [n, \infty)}\{A_k\} \end{cases}. \end{equation} Now the $\displaystyle\sup_{k\in [n,\infty)} \{A_k\}$ and $\displaystyle\inf_{k\in [n,\infty)} \{A_k\}$ are defined as follows: