mt - MATH 1313 (Winter 2003, Lecture 1) Instructor: Roberto...

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Unformatted text preview: MATH 1313 (Winter 2003, Lecture 1) Instructor: Roberto Schonmann Midterm Exam Last Name: First and Middle Names: Signature: UCLA id number (if you are an extension student, say so): Provide the information asked above and write your name on the top of each page using a pen. You should show your work and explain what you are doing; this is more important than just finding the right answer. You can use the blank pages as scratch paper or if you need space to finish the solution to a question. Please, make clear what your solution and answer to each problem is. When you continue on another page indicate this clearly. You are not allowed to sit next to students With Whom you have been studying for this exam or to your friends. Good Luck ! Score 1) (10 points) Let {(xn,yn)} be a sequence of points in a rectangle S = [a, b] X [0, d]. Prove that {(xn,yn)} has a subsequence that converges to a point of S. [Hintz use the corollary to the Bolzano-Weirstrass theorem which states that any sequence {Zn} of real numbers in a finite closed interval [7", s] has a subsequence that converges to a point of [7", W M k aw 3 {xmkg Age. ’14“ —-> 7:. , 5% me («767. CD CM, ) “'5 0st 6. Calailegou, fi 2) (10 points) Suppose that f and g are @discontinuous functions from [R to IR, and that {fn} is a sequence of functions from 1R to R such that fn ——> f pointwise. Can you conclude that g fn —+ gf pointwise? Prove your answer. [Nokcredit will be given for a right answer given for the wrong reason] HM 8C7») QC») -——-> £093 ’v‘zetfi. Una," a... Wm; 4:4, I : am—sa. m) mom-mat). 3) (10 points) Prove the theorem that states that if {fn} is a sequence of continuous functions on [a, b] and fn —> f uniformly, then 4" fn(m)dac ‘—> b mm. [Recall that this is a theorem proved in Section 5.2.] gawk a,.>¢ 3N 3,75, MN a...) {wa-gmls 3-» was? L L b lav-*4.» =5? gfimmlz “Q€W>Jw\g \St£wtw3—%Lm>)&w\ N p o. i 3: s l“ E3 4) (10 points) For f E C[a, b], define Hle = ff|f(x)|da:. Show that - “1 satisfies the triangle inequality v ||f+gl|1 S ||f||1+ llgll1- 5) (10 points) Suppose that (M, p) is a metric space with the property that for all 117,3; 6 M, p(a:,y) E {0, 1, 2,3, Show that if is a sequence of points in M which converges to a point a: E M, then there exists N such that nZN => mnzx. T0)“- 8’3‘ y“). . xvi-427;.) 96, M; N m> P(%)9L3 as: say: Q) :2; Pam») .-.-: 0 CD What. é {DHJZNM} . ...
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This note was uploaded on 09/19/2011 for the course MATH 131B taught by Professor Hitrik during the Winter '08 term at UCLA.

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mt - MATH 1313 (Winter 2003, Lecture 1) Instructor: Roberto...

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