# 0.999.../The limit of a sequence

< 0.999...

In calculus, sequences such as a_{1}=1/1, a_{2}=1/2, a_{3}=1/3..... are discussed. However, most mathematicians and the majority of the mathematical laity prefer the notation **a**_{n}=1/n (**n**≥1) for the sequence above. Often, the limit of a sequence is discussed. A sequence a is said to converge to a limit L if a becomes arbitrarily close to L and stays there. For the case of 0.9999..., the sequence would be a_{n}=1-10^{-n},where n is the number of 9's after the decimal point. For infinitely continuing digits, as n→∞, a_{n}=1-10^{-∞}, thus proving that 0.999999999... until infinity tends to 1.