Let S ⇢ R be a nonempty set which is bounded above,...

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LAST NAME, First Name Homework # 1 ID # : Math 3A03 DUE: by 11:00pm on Saturday January 25, using Crowdmark. 1. Let S R be a nonempty set which is bounded above, and u = sup S . (a) Show that for each n 2 N there exists x n 2 S such that: u - 1 n < x n u, 8 n 2 N . ANSWERS
(b) Prove that the sequence ( x n ) n 2 N converges to u .
Homework # 1 / Math 3A3 -2- NAME: ID #: 2. Let ( a n ) n 2 N be the sequence with a n = p n 2 + 4 - n , n 2 N . (a) Prove that lim n !1 a n = 0 (b) Let b n = n a n . Find L = lim n !1 b n , and prove the limit exists, by using the definition. Continues on page 3. . . ANSWERS u 4 m2 an dn tn E r m In so O an E Z Vn By Squeeze for by def given Go let K E we have an 0 bn man g n TIE 2 when n is large Claim that Le ftp.dn 2 Verify

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