1. ## infinite decimal expansion

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

2. Originally Posted by Kiwili49
How to I show that every infinite string of decimal digits corresponds to a unique real number $\beta$ using the completeness axiom? I am suppose to somehow construct a bounded set with supremum $\beta$ 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 $\beta\in\mathbb{R}$ there is an "infinite" decimal expansion for $\beta$? If you're counting things like $.1\overline{0}$ then you're answer lies in considering a sequence of rational numbers which converges to $\beta$

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

I can't say much more from there.