Real numbers/Sequences/Bounded increasing/Convergence/Decimal expansion/Section

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A bounded and monotone sequence in converges.

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.