Real Numbers and Cauchy Sequences of Rationals - Math Foundations 111

Explore the concept of Cauchy sequences of rational numbers and their role in constructing real numbers in this mathematics lecture. Delve into the logical challenges surrounding Cauchy sequences, including the use of epsilons and N's, and examine how they relate to the notion of limits. Investigate the idea of defining real numbers as limits of Cauchy sequences of rationals, and consider the complexities involved in establishing equivalence between different sequences. Analyze the implications of this approach for computations with real numbers and discuss its significance in modern pure mathematics. Learn about sequences with limits, the proof of Cauchy sequences, and the concept of equivalence classes of Cauchy sequences.

Intro to "real numbers" as Cauchy sequences of rationals
"Sequences with limits are Cauchy sequence"
The usual proof of the previous "theorem"
Cauchy sequences ought to have a limit!
Cauchy sequences of rational numbers
Sequences in the same direction
Equivalence class of Cauchy sequences