4130sols3 - 4130 HOMEWORK 3 Due Thursday February 18 (1)...

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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 . Let ( * ) be the condition: for all ε > 0 there exists N N such that for all m,n > N , | x m - y n | < ε . We need to prove that ( * ) holds if and only if: for all ε > 0 there exists N N such that for all n > N , | x n - y n | < ε . Clearly, ( * ) implies the second condition. So we just need to check that if { x n } ∼ { y n } then ( * ) holds. So suppose { x n } ∼ { y n } . Then let ε > 0. Choose N 1 N such that if n > N , then | x n - y n | < ε/ 2. Since { y n } is a Cauchy sequence, we may also choose N 2 N such that if m,n > N 2 then | y m - y n | < ε/ 2. Now suppose m,n > max { N 1 ,N 2 } . Then | x m - y n | ≤ | x m - y m | + | y m - y n | < ε as required. (2) (a) Using the formula for the partial sums of a geometric series, or otherwise, check
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4130sols3 - 4130 HOMEWORK 3 Due Thursday February 18 (1)...

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