Real numbers/Sequences/Bounded increasing/Convergence/Decimal expansion/Section
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Corollary
Proof
Due to the condition, the sequence is increasing and bounded from above or decreasing and bounded from below. Because of fact, we have a Cauchy sequence which converges in .
This statement is also the reason that any decimal expansion defines a real number. An (infinite) decimal expansion
with (we restrict to nonnegative numbers) and is just the sequence of rational numbers
This sequence is increasing. It is also bounded, e.g. by , so that it defines a Cauchy sequence and thus a real number.