Problem

Source: Iranian National Olympiad (3rd Round) 2007

Tags: induction, limit, real analysis, real analysis unsolved



a) Let $ n_{1},n_{2},\dots$ be a sequence of natural number such that $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$ be a sequence such that $ \epsilon_{i}\in\{1,2\}$. Prove that the sequence: \[ \sqrt[n_{1}]{\epsilon_{1}+\sqrt[n_{2}]{\epsilon_{2}+\dots+\sqrt[n_{k}]{\epsilon_{k}}}}\]is convergent and its limit is in $ (1,2]$. Define $ \sqrt[n_{1}]{\epsilon_{1}+\sqrt[n_{2}]{\epsilon_{2}+\dots}}$ to be this limit. b) Prove that for each $ x\in(1,2]$ there exist sequences $ n_{1},n_{2},\dots\in\mathbb N$ and $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$, such that $ n_{i}\geq2$ and $ \epsilon_{i}\in\{1,2\}$, and $ x=\sqrt[n_{1}]{\epsilon_{1}+\sqrt[n_{2}]{\epsilon_{2}+\dots}}$