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Prove that positive real numbers correspond bijectively.

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Prove that positive real numbers correspond bijectively to decimal expansions not terminating in an infinite string of 9's as follows. The decimal expansion of is where N is the largest integer smaller than is the largest integer is the largest integer and so on.

(a) Show that each is a digit between 0 and 9.
(b) show that there is an n such that n does not equal 9
(c) Conversely, show that for each such expansion ... not terminating in an infinite string of 9's, the set ...]

is bounded and its least upper bound is a real number x with decimal expansion ...

(d) Repeat the exercise with a general base in place of 10

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It is proven that positive real numbers correspond bijectively under a stated set of conditions. The solution is detailed and well-presented.

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