Q:

Consider the infinite expansion of

Denote by the truncation of the above representation after the first decimal digits. So ; for example , , , and so on.

(a) Consider the indexed family of sets , where , open intervals of . Find and . Justify your answers.

b) Consider the indexed family of sets , where , open intervals of . Find and . Justify your answers.

A:

a) and .

b) and .

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?

I didn't really get this stuff down in my foundations course and now its my first HW question for real analysis course .