Exercise · ZW-DENSITY-OF-THE-RATIONALS-010 · ★★★ Advanced
Let \(S=\{q\in\mathbb Q: q^2<2\}\). Which statement is true?
Choose one option.
Show answer
- B. In \(\mathbb R\), \(\sup S=\sqrt2\), which is not rational
Explanation
\(S\) is bounded (by \(2\)) and, by density, its elements approach \(\sqrt2\) without reaching it: \(\sup S=\sqrt2\notin\mathbb Q\). In \(\mathbb Q\) there is no supremum.
Review the concept: Density of the rationals
Property of the rationals \(\mathbb Q\) inside the reals: between any two distinct reals \(a<b\) there is always a rational \(q\).
\[a
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