Problem 150
Prove that if an ordered field satisfies the completeness theorem, then the Archimedean axiom holds. Difficulty: 6. |
Problem 151
Prove that if an ordered field satisfies the completeness theorem, then the Cantor axiom holds. Difficulty: 6. |
Problem 149
Does the ordered field of the rational functions satisfy the completeness theorem: all non-empty set has a supremum? Difficulty: 7. Solution is available for this problem. |
Problem 167
Define recursively the sequence \(\displaystyle x_{n+1}=x_n\left(x_n+\frac1n\right)\) for any \(\displaystyle x_1\). Show that there is exactly one \(\displaystyle x_1\) for which \(\displaystyle 0<x_n<x_{n+1}<1\) for any \(\displaystyle n\). (IMO 1985/6) Difficulty: 9. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |