Real numbers/Cauchy sequence/Monotonic/Introduction/Section

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A problem with the concept of convergence is that in its very formulation already the limit is used, which in many cases is not known in advance. The Babylonian method to construct a sequence (for the computation of say) starting with a rational number gives a sequence of rational numbers. If we consider this sequence inside the real numbers where exists, this sequence converges. However, within the rational numbers, this sequence does not converge. We would like to formulate within the rational numbers alone the property that the members of the sequence are getting closer and closer without referring to a limit point. This purpose fulfills the notion of Cauchy sequence.

Augustin Louis Cauchy (1789-1857)

A real sequence is called a Cauchy sequence, if the following condition holds.

For every , there exists an , such that for all , the estimate


Let be a convergent sequence with limit . Let be given. We apply the convergence property for . Therefore there exists an with

For arbitrary we then have due to the triangle inequality

  Hence we have a Cauchy sequence.

Let be a real sequence. For any strictly increasing mapping , the sequence

is called a subsequence of the sequence.

A real sequence is called increasing, if holds for all , and strictly increasing, if holds for all .

A sequence is called decreasing if holds for all , and strictly decreasing, if holds for all


Let be a real increasing sequence which is bounded from above. Then is a Cauchy sequence.

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.