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Fall 2001 - Fitzsimmons' Class - Exam 2

# Fall 2001 - Fitzsimmons' Class - Exam 2 - Math 142A(P...

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Math 142A (P. Fitzsimmons) Second Midterm Solutions November 14, 2001 1. (a) A real number x is a cluster point of a sequence { a n } if for each > 0 there are infinitely many indices n such that | x a n | < . (b) The n th partial sum of k =1 a k is the finite sum s n = n k =1 a k . (c) The composition f g of two functions f and g with respective domains D f and D g is the function defined by the formula ( f g )( x ) = f ( g ( x )) on the domain D f g consisting of those real numbers x D g for which g ( x ) D f (so that f ( g ( x )) is well defined). 2. Let { a n } n =1 be a Cauchy sequence of real numbers. Let f : R R be a function such that | f ( x ) f ( y ) | ≤ | x y | for all real x and y . Prove that the sequence { f ( a n ) } n =1 is also a Cauchy sequence. Solution. Let > 0 be given. Because { a n } is a Cauchy sequence, there is a cutoff N = N ( ) such that | a m a n | < if m,n > N . But then by the assumed property of f , we have | f ( a m ) f ( a n ) | ≤ | a m a n | < if m,n > N.

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Fall 2001 - Fitzsimmons' Class - Exam 2 - Math 142A(P...

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