Hi Geetha,
The cartesian product of a countably infinite collection of countably infinite sets is uncountable. Let N to be the set of positive integers and consider the cartesian product of countably many copies of N. This is the set S of sequences of positive integers. I am going to show that S is uncountable using a proof by contradiction. I am going to assume that S is countable and hence can be put into a onetoone correspondence with N and then use this fact to produce an member of S that is outside this correspondence. So here is my proof.
Suppose that S is countable then S can be written
S = {s_{1}, s_{2}, s_{3},··· }
that is there is a onetoone correspondence between S and N = {1, 2, 3, ··· }.
At this point the notation get a little cumbersome. Each s_{n} is a sequence of positive integers and hence I am going to write
s_{n} = {x_{n,1}, x_{n,2}, x_{n,3}, ··· }
Thus for each x the first subscript tells you which sequence it is in and the second subscript tells you which element of the sequence it is.
Now I am going to construct a sequence t of positive integers that is not any of the terms s_{n} in S and hence have my contradiction.
If x_{1,1} = 5 then t_{1} = 0 and if x_{1,1} ≠ 5 then t_{1} = x_{1,1}
If x_{2,2} = 5 then t_{2} = 0 and if x_{2,2} ≠ 5 then t_{2} = x_{2,2} , etc.
In general
If x_{k,k} = 5 then t_{k} = 0 and if x_{k,k} ≠ 5 then t_{k} = x_{k,k}
t is a sequence of positive integers and for each positive integer n, t ≠ s_{n} since t_{n} ≠ s_{n,n}
Thus S is not countable.
Penny
