Mid-1_solution

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online