OK: putting the bits together, try this: 

 :In mathematics, the repeating decimal 0.999… (which may also be written as  or ) denotes a real number equal to one. In other words: the notations 0.999… and 1 represent the same real number. The fact that 0.999… equals 1 is well known for being counterintuitive. This is because at first glance, 0.999… appears to suggest a value that is infinitely close to 1, yet somehow falls short of it; in other words, 1 minus an infinitesimal. 
 :However, this is not the case. The equality is based on the definition of a repeating decimal as a limit of a sequence of real numbers, which is itself a real number, without any need to consider infinitesimal quantities. Proofs have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.