Name: geetha Who is asking: Parent Level of the question: All Question: Is the cartesian product of a countably infinite collection of countably infinite sets countable infinite ? and how to prove this? 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 one-to-one 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 = {s1, s2, s3,··· } that is there is a one-to-one correspondence between S and N = {1, 2, 3, ··· }. At this point the notation get a little cumbersome. Each sn is a sequence of positive integers and hence I am going to write sn = {xn,1, xn,2, xn,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 sn in S and hence have my contradiction. If x1,1 = 5 then t1 = 0 and if x1,1 ≠ 5 then t1 = x1,1 If x2,2 = 5 then t2 = 0 and if x2,2 ≠ 5 then t2 = x2,2 , etc. In general If xk,k = 5 then tk = 0 and if xk,k ≠ 5 then tk = xk,k t is a sequence of positive integers and for each positive integer n, t ≠ sn since tn ≠ sn,n Thus S is not countable. Penny