英语翻译
1.Show that for any sequencex 0,x 1 ,x 2 …….of distinet reals there is a sequence a0,b0,a1,b1,a,b2… …of the type deseribed above.
2.Give a rule for continuing the sequence.1/2,1/3,1/4,3/4,1/5,2/5,3/5,4/5,… that will include all rationals in(0,1).
3.Construct a linear fraetional ,apping of(0,1) onto the positive reals.Use this or another method to conclude that the set of positive rationals is countable.How can one then conelude that the set of all rationals is countable?
4.The words on a fixed finite alphabet can be enumerated by listing first the one-letter words,then the weo-letter words,and so on.Use this observation to show that the set of polynomial equations with integer coefficients is countable and hence that the set of algebraic numbers is countable.