# infinite decimal expansion

#### Kiwili49

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
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.