The Real Numbers

Mathematical Analysis

Chapter 1

The notes from Chapter 1 of Katznelson and Katznelson (2024) apply here.

There are two principled ways to introduce the real numbers:

  1. Begin with the Peano axioms and construct \(\symbb{N}\), \(\symbb{Z}\), \(\symbb{Q}\), and then \(\symbb{R}\).
  2. Take as axiomatic \(\symbb{R}\) as a complete, ordered field, and define \(\symbb{N}\), \(\symbb{Z}\) and \(\symbb{Q}\) as subsets thereof.

Schröder (2008) takes the latter approach, which has the virtue of leading more quickly to analysis proper.

Notes

Exercises

  • 1-30: <span class="math inline">\(\symbb{R}\)</span> is unique up to ismorphism

References

Katznelson, Yitzhak, and Yonatan Katznelson. 2024. An Introduction to Real Analysis. Pure and Applied Undergraduate Texts 65. American Mathematical Society.
Schröder, Bernd S. W. 2008. Mathematical Analysis: A Concise Introduction. John Wiley & Sons.