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# Proof of the 0.999... = 1 equality

D
uring the discussion in one of the threads on the FASM forum appeared interesting problem: one of the forumers was sure that inequality $0.(9) = 0.9999 \ldots < 1$ is true. Many people also could agree with him but in this case their intuition would be wrong. Here I decided to show the proof of the equality $0.(9) = 1$ because it's elementary and this equality is an example of paradox between the formal science and human intuitive understanding of mathematics and the world of numbers.

Theorem $0.(9) = 0.9999 \ldots = 1$

Proof

It is clear that $0.(9) = 0.9999 \ldots = 0.9 + 0.09 + 0.009 + \cdots = \sum_{i=1}^{\infty}0.9\frac{1}{10^{i-1}}$ Hence $0.(9)$ is a sum of terms of infinite geometric sequence $\frac{9}{10}, \frac{9}{10^{2}}, \frac{9}{10^{3}}, \cdots$ with the initial value equal to $\frac{9}{10}$ and common ratio $\frac{1}{10}$. Because sum of terms of infinite geometric sequence with the initial value $a$ and common ratio $|r| < 1$ is given by the following formula $\sum_{i=1}^{\infty} ar^{i-1} = \frac{a}{1-r}$ therefore $0.(9) = \sum_{i=1}^{\infty}0.9\frac{1}{10^{i-1}} = \frac{\frac{9}{10}}{1-\frac{1}{10}} = \frac{\frac{9}{10}}{\frac{9}{10}} = 1$ Q.E.D.

© 2007-2014, Mikołaj Hajduk