infinite decimal expansion

Oct 2009
15
0
How to I show that every infinite string of decimal digits corresponds to a unique real number \(\displaystyle \beta\) using the completeness axiom? I am suppose to somehow construct a bounded set with supremum \(\displaystyle \beta\) but do not know how..this problem seems to be way beyond me.
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
How to I show that every infinite string of decimal digits corresponds to a unique real number \(\displaystyle \beta\) using the completeness axiom? I am suppose to somehow construct a bounded set with supremum \(\displaystyle \beta\) but do not know how..this problem seems to be way beyond me.
I don't understand the question.

Are you trying to show that if \(\displaystyle \beta\in\mathbb{R}\) there is an "infinite" decimal expansion for \(\displaystyle \beta\)? If you're counting things like \(\displaystyle .1\overline{0}\) then you're answer lies in considering a sequence of rational numbers which converges to \(\displaystyle \beta\)

If you are given an infinite decimal expansion \(\displaystyle \sum_{n=-N}^{\infty}\frac{a_n}{10^n}\) it clearly (I hope (Worried)) represents a real number and if \(\displaystyle \beta=\sum_{n=-N}^{\infty}\frac{a_n}{10^n},\gamma=\sum_{n=-N}^\infty \frac{a_n}{10^n}\) then (foregoing convergence issues for now) \(\displaystyle \beta-\gamma=\sum_{n=0}^{\infty}0=0\)

I can't say much more from there.