320final

320final - Stony Brook University MAT 320 Introduction to...

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Stony Brook University MAT 320 Introduction to Analysis Final Examination with Solutions December 21, 2007 Work any five problems. Tell us clearly which ones you have chosen. 1. Show how the Bolzano-Weierstrass Theorem (Every bounded sequence contains a con- vergent subsequence) follows from the Least Upper Bound Axiom (Every bounded set of real numbers has a least upper bound). (Solution) This is from the book. You need to show first that every sequence contains a monotone subsequence. Then show that a bounded monotone sequence converges. 2. Prove: If f : R R is positive, bounded away from zero (i.e. K > 0 such that | f ( x ) | ≥ K for all x ) and differentiable, then g ( x ) = q f ( x ) is also differentiable. (Solution) The Chain rule tells us that if f is differentiable at x and if H is differentiable at f ( x ) then H f is differentiable at x (and in fact ( H f ) 0 ( x ) = H 0 ( f ( x )) f 0 ( x )). Here H is the square-root function, which is differentiable at all x > 0. (“Bounded away from zero” was not necessary for this argument.) 3. Prove: if f 0 is continuous and f 0 ( x ) > 0, then f is increasing on a neighborhood of x . Show by example that the hypothesis “ f 0 is continuous” is necessary.

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