Q10.1.1The decimal representation of 0.99999...
The decimal representation of some real numbers is not unique. For example, prove mathematically that \(0.\dot{9} \equiv 0.9999\cdots \equiv 1\).
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Let \(x = 0.9999\cdots\) which, by definition is
\[x = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots\]
This series clearly converges. Then,
\[10x = 9 + \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots = 9 + x \; \Rightarrow \; 9x = 9\; \Rightarrow \; x = 1.\]