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Prove that positive real numbers correspond bijectively to decimal expansions no terminating in an infinite string of 9's as follows. The decimal expansion of $x \in R$ is $N.x_1x_2...$, where $N$ is the largest integer smaller than $x , x_1$ is the largest integer $\leq 10(x-N) , x_2$ is the largest integer $\leq 100(x-(N + x_1 /10 ))$ and so on.

a)Show that each $x_k$ is a digit between 0 and 9.

b) Show that for each $k$ there is an $ \lambda > k$ such that $x_\lambda \neq 9$

c) Conversely show that for each such expansion $N.x_1.x_2...$ not terminating in an infinite string of 9's the set ${\lbrace N, N+ \frac{x_1}{10}, N +\frac{x_1}{10}+\frac{x_2}{100}+...\rbrace}$

d) Repeat with a general base in place of 10.

I really don't know where to build on in this question. I know it is obvious that assuming we are working in base 10 that the division of any number is going to be the fundamental digits from 0-9.

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