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POSITIVE NTH ROOTS
Here we prove, using the least upper bound property ofR
, that positive nth roots exist.The proof is taken from Walter Rudins book Principles of mathematical analysis.
Theorem. Let x > 0 be a positive real number. For every natural number n 1, thereexists a unique positiventh root ofx, which is to sayy >0 R such thatyn =x.
Proof. Givenxwe construct a set Swhose supremum will be the desired element y. Let
S=
t Rt >0, tn x
.
First we claim that Sis nonempty. To see this, considert = x/(x + 1).Sincet 1, it follows thatrn r > x
so r /S, and is an upper bound.By the least upper bound property ofR(which is ultimately equivalent to completeness),
Shas a supremum. Lety= sup(S).
We now claim that yn =x. To prove this, we will derive contradictions from the other twopossibilities, that yn > xandyn < x.Both steps use the following estimate
(1) 0< a < b = bn an
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and note that
0< k < yn
nyn1 =
y
n< y.
Invoking the formula (1) again with a= y k (which weve just shown to be positive) andb= y, we obtain the estimate
yn (y k)n < k n yn1 =(yn
x) n yn1
nyn1 =yn x.
Cancelling the yn terms and multiplying by1, it follows that x x and yn < x, it follows that yn =x.
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