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: $$\nonumber \lim_{n\to \infty} A_n = \begin{cases} \displaystyle\bigcup_{k=1}^\infty A_k&\text{if}\;\{A_k\}\;\text{is expanding}\\[0.3cm] \displaystyle\bigcap_{k=1}^\infty A_k&\text{if}\;\{A_k\}\;\text{is contracting} \end{cases}.$$ Prove the above equation. SolutionProof. If $\{A_k\}$ is either expanding or contracting, then for an infinite sequence \$A_1,A_2,\c…