• Assume a one-to-one correspondence exists between real numbers and rational numbers and try to find a contradiction.
• This would mean a one-to-one correspondence would exist between real numbers and non-negative integers.
• In other words, we would be able to list the real numbers: real number 1, real number 2, real number 3, etc.
• Now think of real numbers as being written in their decimal expansions.
• Remember, we are assuming that every real number is in our list, so to find a contradiction we must "build" a real number which is different from every number in our list.
• For two decimal expansions to be different they need only differ in one position.

That's all the help I'm going to give. I will, however, point out one issue which you will need to think about after you've completing the proof I've outlined above.

Why does this not invalidate the proof I've outlined?