This is true. 0.99999.... repeating to infinity is not really really really close to 1.0. It equals 1.0.
0.9…, or in a variety of other variants such as 0.9, 0.(9)... denotes a real number that can be shown to be the number one. In other words, the symbols "0.999…" and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.
There are a variety of ways to demonstrate this to nonbelievers.
The most easily understood is to revert to other familiar repeating digits. Everyone knows that 1/3 is 0.333... and that 2/3 is 0.666... If you add them together, you get 3/3, which is one.
But now note that the sum of the decimals on the right side of the equation is 0.999...
Therefore, one is equal to (not close to) .999...
You don't agree? Then try this. Subtract .999... from one. What you have is 0.000... An infinitely long string of zeroes, which can only be equal to zero. And if the subtraction of .999... from one leaves zero, then the .999... must be one. But, you say, there's a one at the end that string of zeroes. No, there isn't, because the string of 9s doesn't end.
Other proofs are offered at the
Wikipedia entry.
