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.