Consider an infinitely long table containing all the possible binary digits,
alternatively you could consider it to contain the binary representation
of all integers.
Each entry in the table is infinitely long, padded with zeros perhaps.
For example the top left hand side of the table might look like this:
1 0 0 1 0 0 0 0 ........
1 0 0 0 0 0 0 0 ........
0 1 0 0 0 1 1 0 ........
1 1 0 1 1 1 0 0 ........
0 0 0 1 0 1 0 1 ........
: : : : : : : : ........
: : : : : : : : ........
Consider forming another infinite sequence of digits by moving
along the diagonal of the above table inverting each digit as we go.
So for the above table, this new sequence would start as follows
0 1 1 0 1 ......
This new sequence has a very disturbing attribute, it isn't already
in the table!.
The new sequence is different from the first sequence in the table
because the first digit is different.
The new sequence is different from the second sequence in the table
because the second digit is different.
etc etc....
The new sequence isn't the same as the n'th item in the table because
the n'th digit is different, ie: the n'th digit was inverted.
In summary
We have a table with all possible binary sequences and yet we can construct
a sequence not in the table!
