0.999.../The limit of a sequence

In calculus, sequences such as a₁=1/1, a₂=1/2, a₃=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.