MATH 320 Midterm: MATH 320 2015 Winter Test 1
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This test has 8 questions on 9 pages, for a total of 80 points. No aids of any kind are allowed, including: docu- ments, cheat sheets, electronic devices of any kind (including calculators, phones, etc. ) Student conduct during examinations: each examination candidate must be prepared to produce, upon the request of the invigilator or examiner, his or her ubccard for identi- Student-no: let a1, a2, . be subsets of a metric space x. Give an example to show that the inclusion in part (b) can be proper. Student-no: let b rk be nonempty and compact, and a bc. The metric on rk is the usual metric d(x, y) = |x y|. Recall that the distance d(a, b) from a to b is de ned by d(a, b) = inf{d(a, b) : b b}. (a) Prove that there exists b b such that d(a, b) = d(a, b). Give a short proof if so, or a counterexample if not.
