Proof vs. Poof
Many, many years ago, when teaching math for the first time to a group of middle schoolers, I mentioned that 0.99999.... = 1. They did not believe me; thus, for the next week, I brought in a different "proof" to justify my claim. It became a fun interactive experience, and they were informally introduced to the idea of "proof."
However, most of them never believed my arguments, with some taking this antistand humorously knowing that I would spend my nights trying to concoct a "proof" they would accept. I learned two things. First, mathematical proofs can be difficult if you cannot access (or believe in) the ideas of limits and infintesimal approximations. And second, without a strong background in mathematics, a "proof" becomes no more that a "poof" (which is a "proof" with the "r" for rigor removed!).
For example, think about the standard argument using middle school techniques:
N = 0.999999....
10 N = 9.99999....
10N  N = 9.99999....  0.99999....
9N = 9
N = 1
Why shouldn't they believe this? They would counter with this argument based on their visual use of gaps between two numbers on a number line:
0.9 < 1
0.99 < 1
0.999 < 1
0.9999 < 1
etc.....
Therefore, 0.99999.... < 1
That is, there is no stage when all of a sudden the gap disappears!
It is difficult to counter this physical "vision," as it troubled great mathematicians for centuries. The names abound: Zeno, Archimedes, Newton, Leibniz, Cauchy, Weierstrass, Kepler, Cavalieri, Wallis, Lagrange, et al. The idea of an infinitesimal and/or limit is not trivial. For example, where does either appear in the two arguments shown above....or does it?
What is your BEST argument for "proving" that 0.9999.... = 1? Or, perhaps you do not believe it yourself. In an attempt perhaps to put this dilemma to bed, Karin Katz and Mikhail Katz published the article "When is 0.9999... Less Than 1?" in The Montana Mathematics Enthusiast (2010). Check it out....though be forewarned that the mathematics is not trivial....and unfortunately not usable with most middle school students.
At the root of all this lies the important ideas of "logical steps in a proof" and overall "belief in a proof." And, for esteemed mathematicians, these ideas often pose difficulties. One needs only remember the mathematicians difficulty in "accepting" the computerbased "proof" of the Four Color Problem!
