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 ²> 0 there are in±nitely many indices n such that | x a n | . (b) The n th partial sum of k =1 a k is the ±nite 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 de±ned 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 de±ned). 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 ²> 0 be given. Because { a n } is a Cauchy sequence, there is a cuto² N = N ( ² ) such that | a m a n | if m,n > N . But then by the assumed property of
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This note was uploaded on 04/23/2008 for the course MATH 142A taught by Professor Fitzsimmons during the Fall '01 term at UCSD.

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

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