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# INTRO CLASSICAL ANALYSIS MATH 615

Texas A&M

GPA 3.92

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This 2 page Study Guide was uploaded by Evert Christiansen on Wednesday October 21, 2015. The Study Guide belongs to MATH 615 at Texas A&M University taught by Natarajan Sivakumar in Fall. Since its upload, it has received 24 views. For similar materials see /class/225999/math-615-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15

Siuakumar M615 080 Example S heet 2 1 Suppose 1 and 52 are bounded subsets of the real line Prove the following statements i 51 U 52 is bounded ii supSl U 52 maxsupSlsupSg iii infSl U 52 mininfSlinfSg D Suppose A and B are nonempty bounded subsets of R and let Ssab LEA bEB Prove the following statements i S is bounded n we mm supltBgt iii infS infA infB 3 Suppose that an is a sequence of nonnegative numbers Shew that lim an 0 if and only if lim sup an O 4 a Suppose 3n and 25 are bounded sequences of real numbers Prove the following state ments i lim supsn tn lim sup 3quot lim sup tn ii lim infsn tn 2 lim inf squot lim inf tn b Find a pair of bounded sequences 3quot and 25 such that lim supsn tn 7 lim sup 3quot lim sup tn 5 Suppose 3 is a sequence of real numbers Prove the following statements i lim 3 00 if and only if lim sup 3 lim inf 3 00 ii lim 3 00 if and only if lim sup 3 lim inf 3n 00 De nition Let 3 be a given sequence of real numbers A real number L is said to be a subsequential limit of 3 if there is some subsequence of 3 which converges to L 6 Suppose 3quot is a bounded sequence of real numbers Let M lim sup 3quot and m lim inf squot TH Prove if L is any subsequential limit of 5n7 then m g L g M Recall from lecture that both M and m are subsequential limits Combining this with the assertion above we nd that the limit superior of a sequence is the largest subsequential limit of the sequence7 whereas the limit inferior is the smallest subsequential limit of the sequence 1 Suppose that an is a sequence of real numbers Prove the following statements i lim sup an 00 if and only if there is a subsequence of an which diverges to 00 i lim inf an 00 if and only if there is a subsequence of an which diverges to 00 8 3 C Suppose an is a sequence of positive numbers Let lim supaiTL L7 L E R Prove the following statements i If 0 g L lt 17 then the in nite series 2 an is convergent n1 ii If L gt 17 then the in nite series 2 an is divergent n1 Let A be a xed positive number and consider the problem of nding the square root of A This problem is tantamount to solving the equation x 07 where x x2 7 A The Newton Raphson method provides a certain algorithm to solve the equation speci cally7 one considers the sequence xn de ned recursively n 90n1390n f 7 n where x1 is a certain speci ed number i Verify that for x x2 7 A7 the recursion above reduces to the following 1 A mn1 n7 7121 ii Let the initial value x1 be chosen such that x1 gt 0 and x gt A Show that the resulting sequence xn is nonincreasing and bounded below iii Show that lim xn xZ Suppose that an is a sequence of real numbers Prove that an admits a monotone subsequence Consider the set I m E N an 2 am V n gt m and discuss the following cases I is in nite7 I is nite or empty ii Use along with other results discussed in lecture to show that every Cauchy sequence of real numbers must be convergent Suppose 3 is a bounded sequence of real numbers Let 731quot 3n 7 7 nEN i Show that an is a bounded sequence ii Prove that lim inf 3 lim inf an lim sup an lim sup 3 iii Deduce the following result from ii If lim 3 L7 then lim an L iv Find a bounded7 divergent sequence 3 such that an is convergent This will demon strate that the converse of iii is false in general

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