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Take any two distinct, real numbers. Any two, doesn't matter which two. Ok, so what makes them distinct? What makes the distinct is the fact that there are an infinite number of real numbers inbetween them. Can you name even one number between .999.. and 1? I can't. If you can, then you've proven they're distinct. If you cannot, then they are not distinct, and are thus equivalent.
Of course, I would have to prove that you cannot, but there are plenty of valid proofs already listed in this thread. It's not "voodoo math".. it is perfectly valid.
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