Do Unique nth Real Roots Exist? (Baby Rudin Proof Explained)

Описание: Study Help for Baby Rudin, Part 1.6. I prove that every positive real number has a unique positive real nth root. The proof requires the Completeness Axiom of the real numbers R, which is the same as the least upper bound property of R. This is an in-depth explanation of the proof in "Principles of Mathematical Analysis", by Walter Rudin: https://amzn.to/39XOqzn

Proof of Existence of nth Real Roots with the Completeness Axiom and a Useful Inequality

After applying the Completeness Axiom to construct the proposed nth root y of x, the proof is by double reductio ad absurdum. We assume y^n is not equal to x. The two cases 1) y^n is less than x and 2) y^n is greater than x both lead to contradictions. Proving these contradictions requires an inequality arising from the factorization of b^n - a^n, the difference of two nth powers. It also requires some tricky problem-solving and scratch work.

#realanalysis #realanalysisproofs #babyrudin

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⏱️TIMESTAMPS⏱️
(0:00) Does the cube root of 10 exist?
(1:18) Want to use Real Analysis (Completeness) to prove this.
(2:56) Strategy for using the Completeness Axiom
(4:00) General existence theorem for positive nth real roots of positive real numbers
(4:45) Overall strategy for the proof
(5:56) Uniqueness proof
(6:24) Construct E whose supremum y is the nth root of x
(6:42) Show E is nonempty and bounded above
(8:06) Prove y^n = x by contradiction (double reductio ad absurdum)
(8:39) Key Lemma (Useful Inequality)
(9:26) Numerical demonstration of the lemma
(9:50) Graphical demonstration of the lemma
(10:28) Hard part of the proof (the "analysis" part)

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