In Stephen Abbott's Understanding Analysis, when proving that the set of rational numbers is dense in the set of real numbers, Abbott picks an integer $m$ such that $$m - 1 \leqslant na < m,$$ where $n$ and $a$ are a natural number and a real number respectively.
Intuitively it makes sense that such an $m$ exists, but it seems to me that Abbott is taking liberties here. How do I know from the fact that the set of reals is a complete ordered field that such an $m$ exists? (Abbott doesn't discuss the real number axioms but I know them from elsewhere.)
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