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4130hw3

# 4130hw3 - number deﬁned by this Cauchy sequence is...

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4130 HOMEWORK 3 Due Thursday February 18 (1) Let { x n } and { y n } be Cauchy sequences of rational numbers. Prove that { x n } ∼ { y n } if and only if for all ε > 0 there exists N N such that for all m,n > N , | x m - y n | < ε . (2) (a) Using the formula for the partial sums of a geometric series, or otherwise, check that the sequence of rational numbers whose n th term is x n = n X i =1 e i 10 i is a Cauchy sequence, for any integers e i with 0 e i 9. (Remark: the real
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Unformatted text preview: number deﬁned by this Cauchy sequence is denoted 0 .e 1 e 2 e 3 ... ) (b) Show that 0 . 999 ... = 1. (3) Section 2.2.4 #5. (4) Using the triangle inequality, show that for any a,b ∈ R , we have || a | - | b || ≤ | a-b | . Using this, show that if { x n } is a sequence of real numbers which converges to L , then {| x n |} converges to | L | . 1...
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