[techno_tsar]

There's a rigorous proof on Wikipedia, but there's simpler ways to show it.

For example, we know that 1/3 = 0.333...

3 * 1/3 = 3 * 0.333...

1 = 0.999...

You can also do it with subtraction. For example, 1 - 0.999... = x. Assuming x is greater than 0, then it should evaluate to 0.000...1.

But we can't have the digit 1 after an infinite number of zeros. If there truly were a "1" after infinite zeros, it implies reaching the end of infinity, which is a logical contradiction. So x can't be greater than 0.