Is there a construction of some subset of real numbers where the existence of multiplicative inverse of real numbers be derived by its other properties?
In particular, I am curious of the following more limited question: consider the set of limits of all convergent rational sequences $S$. For every $x\in S$, are we guaranteed to find a sequence in $S$ whose limit is the multiplicative inverse of $x$? (Where multiplication is defined via constructing rationals as equivalent classes of ordered pairs of integers.)