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.
Well-said. So I take if you'd say it follows from this that if the term in question doesn't fall into the realm of real numbers, then this type of proof can't necessarily be applied?
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Tebow to Jests?
http://24.media.tumblr.com/tumblr_m19b15IB731qzdb47o1_400.png