Completion Notes

If n the rst term is smaller than by denition xn xm

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Unformatted text preview: 1 ⇒ x − x 1 V < 1. Pick x1 = x 1 . ∈IN (2) x (3) ∈IN Cauchy ⇒ ∃ > 1 s.t. ≥ ∈IN September 7, 2011 3 > 2 s.t. ≥ 2 (2) ⇒x (3) (2) −x 2 (3) (2) V < 1 . Pick x2 = x 2 1 . 3 2 . (3) ⇒ x −x 3 V < Pick x3 = x 3 . . . . (n) (n) (n) (n) 1 x Cauchy ⇒ ∃ n > n−1 s.t. ≥ n ⇒ x − x n V < n . Pick xn = x n . ∈IN . . . (n) In the example sketched below, the x n ’s are circled (in the simplest case in which n happens to be n). x Cauchy ⇒ ∃ 2 3 Completion 4 X(1) X(3) · · · X(2) (1) x4 (1) x3 (1) x2 (1) x1 (2) x4 (2) x3 (2) x2 (2) x1 Proof that {xn }n∈IN is Cauchy: Let ε > 0. By the triangle inequality xn − xm (n) V ≤x n (n) =x n (n) (n) −x V (n) −x V +x ( m) (n) x + ( m) −x V +x ( m) ( m) −x (n) −x V −X −X + X(n) − X(m) for any ∈ IN. ◦ If ≥ n , the first term is smaller than ◦ By definition, X(n) − X(m) H = lim 1 n. (n) x →∞ ( m) −x V V ( m) m (2) H ( m) H +x ( m) −x m V . So there is a natural number Nn,m ε such that the second term is s...
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