Originally Posted by

**Danneedshelp** Q:

Consider the infinite expansion of $\displaystyle \sqrt{2}$ $\displaystyle =$ $\displaystyle

1.414213562373095048801688724209698078569671875376 9

• • • = 1.x_{1} x_{2} . . . x_{n} . . . . $

Denote by $\displaystyle r_{n}$ the truncation of the above representation after the first $\displaystyle n$ decimal digits. So $\displaystyle r_{n} = 1.x_{1}x_{2} . . . x_{n}$ ; for example $\displaystyle r_{1} = 1.4$, $\displaystyle r_{2} = 1.41$, $\displaystyle r_{3} = 1.141$, and so on.

(a) Consider the indexed family of sets $\displaystyle \{A_{n}\}_{n\in\mathbb{N}}$, where $\displaystyle A_{n} = (-\infty, r_{n} )$, open intervals of $\displaystyle \mathbb{R}$. Find $\displaystyle \bigcup_{n\in{\mathbb{N}}}A_{n}$ and $\displaystyle \bigcap_{n\in{\mathbb{N}}}A_{n}$. Justify your answers.

b) Consider the indexed family of sets $\displaystyle \{B_{n}\}_{n\in\mathbb{N}}$, where $\displaystyle A_{n} = [\frac{1}{n}, r_{n} )$, open intervals of $\displaystyle \mathbb{R}$. Find $\displaystyle \bigcup_{n\in{\mathbb{N}}}B_{n}$ and $\displaystyle \bigcap_{n\in{\mathbb{N}}}B_{n}$. Justify your answers.

A:

a) $\displaystyle \bigcup_{n\in{\mathbb{N}}}A_{n}=(-\infty,\sqrt{2})$ and $\displaystyle \bigcap_{n\in{\mathbb{N}}}A_{n}=(-\infty,1.4]$.

b) $\displaystyle \bigcup_{n\in{\mathbb{N}}}B_{n}=(0,\sqrt{2})$ and $\displaystyle \bigcap_{n\in{\mathbb{N}}}B_{n}=[1,1.4)$.

I do not know how to prove my answers. I have tried, and can't seem to reach a conclusion. Could someone please help me with either a or b? Is there a systematic whay to approach these kinds of problems? How do I use the Archimedean principle?