# Real numbers correspond bijectively to decimal expansions

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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