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 .
.
.
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(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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This note was uploaded on 02/13/2014 for the course MATH 511 taught by Professor Joelfeldman during the Spring '13 term at UBC.
 Spring '13
 JoelFeldman

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