Cantor's Diagonalization Argument proves that the real numbers are uncountable. Even if you list infinitely many real numbers between 0 and 1, we can always construct a new number that is not on your list.
The Method: Take the $n$-th digit of the $n$-th number and flip it (0 → 1, 1 → 0). The result differs from every number in the list by at least one digit. Instructions: Scroll the list of random binary numbers. Generating the "Diagonal Number" creates a sequence guaranteed to be missing from the list.
New Number (The Counter-Example)
0....
This number differs from Row $N$ at position $N$.