final - MAT125A Final and Solutions March 17, 2010 1. Show...

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Unformatted text preview: MAT125A Final and Solutions March 17, 2010 1. Show that if f : [0 , ) R is continuous and lim x f ( x ) is finite, then f is bounded on [0 , ). Solution: Let L = lim x f ( x ). Then there exists > 0 such that x > implies f ( x ) < L + 1. Since f is continuous and [0 , ] is compact, f is bounded on [0 , ] say by M . Let = max { L + 1 ,M } . Then | f ( x ) | for all x 0. 2. Prove that the metric space ( X,d ) where X = R and d : X X [0 , ) defined by d ( x,y ) = ( x = y 1 x 6 = y is complete. That is, show that every Cauchy sequence { x n } in ( X,d ) converges to a point x X . Solution: Let { x n } be a Cauchy sequence in X . That means that for all > there exists N such that d ( x n ,x m ) < for n,m > N . In particular, there exists an N such that d ( x n ,x m ) < 1 / 2 for all n,m > N . But, by the definition of the metric, d ( x n ,x m ) < 1 implies x n = x m . So the sequence { x n } is constant for n > N and hence convergent....
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This note was uploaded on 03/18/2012 for the course MAT MAT 125A taught by Professor Bremer during the Winter '10 term at UC Davis.

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final - MAT125A Final and Solutions March 17, 2010 1. Show...

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