Exercise - sheet 01

[LeGiang]

xin mời các cao thủ ra tay:

Let [IMG]http://img359.images****.us/img359/5933/01up6.gif[/IMG] be an integer. Show that a real number x is rational if and only if the sequence [IMG]http://img485.images****.us/img485/3466/01mq1.gif[/IMG] of digits of x in the expansion in basis b [IMG]http://img412.images****.us/img412/7196/01rp6.gif[/IMG]
is ultimately periodic!

Let [IMG]http://img359.images****.us/img359/5933/01up6.gif[/IMG] be an integer. Let [IMG]http://img469.images****.us/img469/7879/01ss2.gif[/IMG] be a bounded sequence of rational integers and [IMG]http://img469.images****.us/img469/2613/01ks9.gif[/IMG] an increasing sequence of positive numbers. Assume there exists c>0 and [IMG]http://img529.images****.us/img529/8151/01co0.gif[/IMG]such that, for all[IMG]http://img465.images****.us/img465/213/01gi4.gif[/IMG], [IMG]http://img507.images****.us/img507/4563/01fj7.gif[/IMG]

a)deduce, for all [IMG]http://img400.images****.us/img400/4476/01lt5.gif[/IMG] and [IMG]http://img465.images****.us/img465/213/01gi4.gif[/IMG], [IMG]http://img209.images****.us/img209/6797/01jb3.gif[/IMG]
b)show that the number [IMG]http://img520.images****.us/img520/6936/01rd7.gif[/IMG] is irrational if and only if the set [IMG]http://img139.images****.us/img139/6198/01wo8.gif[/IMG] is infinite!