Real numbers/Bounded monotonic increasing sequence/Cauchy sequence/Fact/Proof

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Let denote a bound from above, so that holds for all . We assume that is not a Cauchy sequence. Then there exists some such that for every , there exist indices fulfilling . Because of the monotonicity, there is also for every an with . Hence, we can define inductively an increasing sequence of natural numbers satisfying

and so on. On the other hand, there exists, due to the axiom of Archimedes, some with

The sum of the first differences of the subsequence , , is

This implies , contradicting the condition that is an upper bound for the sequence.