#Math101 Session 35

Time：Mar. 19, 2026, 15:30-17:30

Location：D212 Hai Tian Building, Chang'an Campus

Speaker：Mr. Yang, Dongyu (Class of 2024)

Title：The completeness of the real numbers

Abstract：We continue our study of the foundational structure of the real number system by focusing on key formulations of completeness. We begin with the axiom of nested intervals, in its weaker form, which states that nested closed intervals have nonempty intersection. We then examine the supremum principle and show that, in an ordered field, the supremum principle is equivalent to the combination of the axiom of nested intervals and the Archimedean property, together capturing the full completeness of the real numbers. Finally, we introduce Dedekind cuts as a rigorous construction of the real numbers from the rational numbers.

（文：申爽/审核：郭千桥）