Mid-1_solution

# Mid-1_solution - Solutions to 131A Midterm 1 Spring 09 Note...

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Solutions to 131A Midterm 1, Spring 09. Note: Proofs must be rigorous. Drawings will not be credited, though allowed. Q1. Let A and B be two subsets of R , both bounded below. Let S = A + B , the set of elements of the form a + b with a A and b B . a). Show that S is bounded below. b). Show that inf ( S ) = inf ( A ) + inf ( B ). Proof. It is dual to the proof of Exercise 4.14 a) which is online (the midterm 1 review). Go read it if you haven’t.

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Q2. Let s n and t n be two convergent sequences. Suppose s n t n for all but ﬁnitely many n . Show that lim ( s n ) lim ( t n ). Proof. Method of contradiction. Put lim ( s n ) = s and lim ( t n ) = t for the sake of simplicity. Suppose the opposite, i.e., s > t . (Imagine a picture that on the x -axis, s is to the right of t .) Let d = s - t be the distance, and put ± = d 3 . WLOG (Without loss of generality), we may assume s n t n is true for all n . See the remark in the very end of this ﬁle (i.e., after Q6 ). For this particular
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Mid-1_solution - Solutions to 131A Midterm 1 Spring 09 Note...

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