From: Westbrick | #011
Except that there's plenty of debate about it. When you treat .999 as a normal number susceptible to the rules of basic algebra, then you can perform those proofs we all learned in middle school about how .999 = 1. The real question boils down to whether or not such equations can be performed on an infinite.
Those proofs aren't proofs for the reasons you stated.
One property of real numbers is that for every two real numbers x and y, either x = y or there exists a real number z such that z is between x and y. Using this property you can show that 0.999~ and 1 are equal.
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No amount of planning will ever replace dumb luck.