Problem 345
The sequence \(\displaystyle a_n\) is monotone and it has a convergent subsequence does it imply that \(\displaystyle a_n\) is convergent? Difficulty: 4. Solution is available for this problem. |
Problem 346
Prove that if \(\displaystyle |a_{n+1} -a_n |\leq 2^{-n}\) for all \(\displaystyle n\), then \(\displaystyle (a_n )\) is convergent. Difficulty: 5. |
Problem 347
Prove that if the Bolzano-Weierstrass theorem holds in an ordered field, then it is isomorphic to \(\displaystyle \R\). Difficulty: 8. |
Problem 348
Prove that if in an Archimedean ordered field every Cauchy sequence is convergent, then every bounded set has a least upper bound. Difficulty: 8. |
Problem 349
Prove that every Cauchy-sequence is convergent, using the one-dimensional Helly-theorem. Difficulty: 8. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |