Problem

Source: Iranian National Olympiad (3rd Round) 2007

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



a) Let n1,n2, be a sequence of natural number such that ni2 and ϵ1,ϵ2, be a sequence such that ϵi{1,2}. Prove that the sequence: n1ϵ1+n2ϵ2++nkϵkis convergent and its limit is in (1,2]. Define n1ϵ1+n2ϵ2+ to be this limit. b) Prove that for each x(1,2] there exist sequences n1,n2,N and ni2 and ϵ1,ϵ2,, such that ni2 and ϵi{1,2}, and x=n1ϵ1+n2ϵ2+