Find the Infimum and supremum of M set

Question

$$M = \bigcap_{n\in N}{I_n},$$ $$I_n = \bigcup_{k = 1}^{3^n - 2} \left(\frac{k}{3^n},\frac{k+1}{3^n}\right)$$

Answer 1

Hint

First prove that if $M=\bigcap_{n\in\Bbb N}I_n$, then $$ \sup M=\inf_n\sup I_n\\ \inf M=\sup_n\inf I_n $$