Exercise · ZW-ARCHIMEDEAN-PROPERTY-007 · ★★☆ Intermediate
In an ordered field there is \(\varepsilon>0\) with \(\varepsilon<\tfrac1n\) for every natural \(n\). Which property fails?
Choose one option.
Show answer
- A. The Archimedean property
Explanation
Such an \(\varepsilon\) satisfies \(n\varepsilon<1\) for all \(n\), so no multiple of it exceeds \(1\). The other ordered field axioms are unaffected.
Review the concept: Archimedean property
The property of \(\mathbb R\) by which, however small a positive number \(x\) is, some multiple of it by a natural \(n\in\mathbb N\) exceeds any real \(y\).
\[x>0,\ y\in\mathbb R\ \Longrightarrow\ \exists\, n\in\mathbb N:\ nx>y\]
More practice problems: archimedean property
- ZW-ARCHIMEDEAN-PROPERTY-001 What does the Archimedean property state?
- ZW-ARCHIMEDEAN-PROPERTY-002 What is the smallest natural n with tfrac1n<0.003?
- ZW-ARCHIMEDEAN-PROPERTY-003 By the Archimedean property, what is the infimum of \1/n: n∈mathbb N\?
- ZW-ARCHIMEDEAN-PROPERTY-004 If x=0.002 and y=5, what is the smallest natural n with nx>y?
- ZW-ARCHIMEDEAN-PROPERTY-005 Which statement is equivalent to the Archimedean property?
- ZW-ARCHIMEDEAN-PROPERTY-006 What is the smallest natural n with (n)/(n+1)>0.999?
- ZW-ARCHIMEDEAN-PROPERTY-008 For x=tfrac37 and y=100, what is the smallest natural n with nx>y?
- ZW-ARCHIMEDEAN-PROPERTY-009 Which is the correct idea for proving the Archimedean property from the completeness axiom?