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hw_w11

# hw_w11 - (a If s n and t n are divergent sequences then s n...

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Mathematics 220 Homework for Week 11 Due April 2 1. 12.2 2. 12.4 3. 12.6 4. Prove that for any x, y R , | x | - | y | ≤ | x - y | . Hint: prove -| x - y | ≤ | x | - | y | ≤ | x - y | . Hint on the hint: The triangle inequality is your friend. 5. For each of the following prove or give a counter-example (a) If { a n } converges to L then {| a n |} converges to | L | . Hint: you need to use the result of Q4 (b) If {| a n |} is convergent then { a n } is convergent. (c) a n 0 if and only if | a n | → 0. 6. Find an example of each of the following (a) A convergent sequence of rational numbers having an irrational limit. (b) A convergent sequence of irrational numbers having a rational limit. 7. For each of the following give a counter-example to show they are false.
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Unformatted text preview: (a) If { s n } and { t n } are divergent sequences then { s n + t n } diverges. (b) If { s n } and { t n } are divergent, then { s n t n } is divergent. (c) If { s n } and { s n t n } are convergent, then { t n } is convergent. 8. Let { a n } be a sequence of positive numbers. Prove that if { a n } diverges to inﬁnity then { 1 /a n } converges to 0. 9. Let { a n } , { b n } , { c n } be sequences such that a n ≤ b n ≤ c n for all n ∈ N . Prove that if a n → b and c n → b then b n → b . Page 1 of 1...
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