This is the explanation given online.But I m not able to understand as to why set of real numbers is mentioned.
we know that the set of real numbers between 0 and 1(
denoted by (0, 1) is uncountable. Let us associate to each real number [0, 1) a function from
the set of positive integers to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} as follows:
If x is a real number whose decimal representation is 0.d 1 d 2 d 3 · · · (with ambiguity resolved
by forbidding the decimal to end with a infinite string of 9’s). The we associate to the x the
function whose rule is given by f (n) = d n .
Clearly, this is a one to one function from [0,1) and a subset of all functions from positive
integers to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Two different real numbers must have different
decimal representation, so the corresponding functions are different.(A few functions are left
out, because of forbidding representations such as 0.2399999 · · ·).
Since (0,1) is uncountable, the subset of functions we have associated with them must be
uncountable. But the set of all such functions has at least this cardinality, so it, too, must
