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# final - MAT125A Final and Solutions 1 Show that if f[0 R is...

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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 α > 0 such that x > α implies f ( x ) < L + 1. Since f is continuous and [0 , α ] is compact, f is bounded on [0 , α ] — say by M . Let γ = max { L + 1 , M } . Then | f ( x ) | ≤ γ 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 ) = ( 0 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 > 0 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. 3. Show that if { a k } is a sequence of positive real numbers such that (1) X k =1 a k < , then (2) X k =1 ln(1 + a k ) < .

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final - MAT125A Final and Solutions 1 Show that if f[0 R is...

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