existence of square roots of non-negative real numbers
Theorem.
Every non-negative real number has a square root.
Proof.
Let x≥0∈ℝ. If x=0 then the result is trivial, so suppose x>0 and define S={y∈ℝ:y>0 and y2<x}. S is nonempty, for if 0<y<min{x,1}, then y2<y<x, and y∈S. S is also bounded above, for if y>max{x,1}, then y2>y>x, so such a y is an upper bound of S. Thus S is nonempty and bounded, and hence has a supremum which we denote L. We will show that L2=x. First suppose L2<x. By the Archimedean Principle there exists some n∈ℕ such that n>(2L+1)/(x-L2). Then we have
(L+1n)2=L2+2Ln+1n2<L2+2Ln+1n<x. | (1) |
So L+1/n is a member of S strictly greater than L, contrary to assumption. Now suppose that L2>x. Again by the Archimedean Principle there exists some n∈ℕ such that 1/n<(L2-x)/2L and 1/n<L. Then we have
(L-1n)2=L2-2Ln+1n2>L2-2Ln>x. | (2) |
But there must exist some y∈S such that L-1/n<y<L, which gives x<(L-1/n)2<y2, so that y∉S, a contradiction. Thus it must be that L2=x.
∎
Title | existence of square roots of non-negative real numbers |
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Canonical name | ExistenceOfSquareRootsOfNonnegativeRealNumbers |
Date of creation | 2013-03-22 16:32:42 |
Last modified on | 2013-03-22 16:32:42 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 8 |
Author | PrimeFan (13766) |
Entry type | Theorem |
Classification | msc 11A25 |
Related topic | AxiomOfAnalysis |
Related topic | ArchimedeanProperty |
Related topic | Supremum |
Related topic | ExistenceOfNthRoot |