Math 2.0

concept

Construction of the reals from Cauchy sequences

Summary

This construction defines the real numbers as limits of sequences of rational numbers. A sequence (xn)(x_n) of rational numbers is a Cauchy sequence when, for each rational ε>0\varepsilon > 0, there is NN with ∣xm−xn∣<ε|x_m - x_n| < \varepsilon for all m,n≥Nm, n \ge N. Two Cauchy sequences (xn)(x_n) and (yn)(y_n) are equivalent when xn−ynx_n - y_n tends to 00: for each rational ε>0\varepsilon > 0, there is NN with ∣xn−yn∣<ε|x_n - y_n| < \varepsilon for all n≥Nn \ge N. A real number is an equivalence class of Cauchy sequences of rational numbers. Addition and multiplication act term by term: [(xn)]+[(yn)]=[(xn+yn)][(x_n)] + [(y_n)] = [(x_n + y_n)] and [(xn)]⋅[(yn)]=[(xnyn)][(x_n)] \cdot [(y_n)] = [(x_n y_n)].

Remarks

The rational number qq corresponds to the class of the constant sequence (q,q,q,… )(q, q, q, \dots). The sequence 1,1.4,1.41,1.414,…1, 1.4, 1.41, 1.414, \dots of decimal approximations represents 2\sqrt{2}. The result is a complete ordered field, with the order on the reals. The same method completes any metric space.

Explanations

L1 human

A real number is something that rational numbers can approximate as well as you like. The construction takes the approximating sequences themselves as the new numbers. Two sequences that get closer and closer to each other count as the same real number.

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Sources

  1. Web https://en.wikipedia.org/wiki/Construction_of_the_real_numbers, Construction from Cauchy sequences

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