DMcG, sometimes I enjoy playing the temporary pedant myself, but not this time. Mysha (who is male, as my private directory of Mudcatters tells me) did not provide a disproof - how could he - but a "disproof", i.e. as precise as your "proof", and therefore pointing at that lack of precision. No need though to mend it, though.

It is basically high school maths, not requiring higher academic studies. Still, regarding Airymouse of 31 Aug 13 - 10:41 AM, it is noteworthy (though not "puzzling") that all numbers that have a finite decimal representation also have another, infinite one (ending with an infinitely many 9s), whereas all other numbers only have one representation.

In contrast, MtheGM's original question, which properly reads "Are there infinitely many numbers with that property, and if not, which ones or how many or which is the largest one?", does not look easy to me at all. A friend of mine who minored in maths gave up after ten minutes. I would not bet that every PhD mathematician performs better.