【英语翻译1.Showthatforanysequencex0,x1,x2…….ofdistinetrealsthereisasequencea0,b0,a1,b1,a,b2……ofthetypedeseribedabove.2.Givearuleforcontinuingthesequence.1/2,1/3,1/4,3/4,1/5,2/5,3/5,4/5,…thatwi】

　　英语翻译

　　1.Showthatforanysequencex0,x1,x2…….ofdistinetrealsthereisasequencea0,b0,a1,b1,a,b2……ofthetypedeseribedabove.

　　2.Givearuleforcontinuingthesequence.1/2,1/3,1/4,3/4,1/5,2/5,3/5,4/5,…thatwillincludeallrationalsin(0,1).

　　3.Constructalinearfraetional,appingof(0,1)ontothepositivereals.Usethisoranothermethodtoconcludethatthesetofpositiverationalsiscountable.Howcanonethenconeludethatthesetofallrationalsiscountable?

　　4.Thewordsonafixedfinitealphabetcanbeenumeratedbylistingfirsttheone-letterwords,thentheweo-letterwords,andsoon.Usethisobservationtoshowthatthesetofpolynomialequationswithintegercoefficientsiscountableandhencethatthesetofalgebraicnumbersiscountable.