Square roots exist, Real Analysis 1

Описание: I review three foundational results that are assumed throughout the course and proved earlier in the Logic and Proof Writing playlist (https://www.youtube.com/playlist?list.... These results appear early in a standard real analysis course and are used repeatedly in both computations and proofs.

I first recall the triangle inequality and explain why absolute values play such a central role in real analysis, particularly when establishing equality via distance arguments. I then briefly discuss the irrationality of the square root of two as a historically important example that motivates the existence of irrational numbers.

The main focus of the video is a complete proof that square roots exist for all non-negative real numbers. Using the supremum property of the real numbers and the Archimedean principle, I construct the square root of a given non-negative number and carefully rule out the alternative cases.

This lecture emphasizes proof structure, logical case analysis, and the role of foundational axioms in constructing familiar objects on the real number line.

#realanalysis #advancedcalculus #mathematicalproofs #supremum #maths #mathematics #realnumbers