Suppose that the output of the machine is a1,a2,a3,... as the
first sequence, b1,b2,b3,... as the second sequence,
c1,c2,c3,... as the third sequence and so on. Now consider
the infinite sequence
which differs from the first sequence in the first place, from the
second sequence in the second place, from the third sequence in
the third place, ... and so on and in general it differs from the
nth sequence in the nth place. So this sequence cannot be one
of the sequences produced by the machine.
Consider the binary numbers written in ascending order: 0, 1, 10,
11, 100, 101, ... etc. This is an infinite set of finite sequences
of zeros and ones and it is written as an ordered list.
By contrast we cannot write in an ordered list all the infinite
sequences in the infinite set of infinite binary sequences.
To prove this fact we suppose that the all the infinite sequences
in the infinite set of infinite binary sequences can be written in
an ordered list and we reach a contradiction. Let
a1,a2,a3,... be the first sequence, b1,b2,b3,... be the
second sequence, c1,c2,c3,... be the third sequence and so
on. Now consider the infinite sequence
which differs from the first sequence in the first place, from the
second sequence in the second place, from the third sequence in
the third place, ... and so on and in general it differs from the
nth sequence in the nth place. So it cannot be one of the
sequences in the list which contradicts our assumption that we
could write an ordered list. Hence no such list exists.