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by: Moshe Swift III

FunctionalAnalysis MATH640

Moshe Swift III
GPA 3.63


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Class Notes
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This 1 page Class Notes was uploaded by Moshe Swift III on Wednesday September 23, 2015. The Class Notes belongs to MATH640 at Drexel University taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/212301/math640-drexel-university in Mathematics (M) at Drexel University.

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Date Created: 09/23/15
Notes on Upper and Lower Limits De ne the extended real line as R R U 700 U Theorem 1 Let an be a sequence of real numbers anda E R Then the following statements are equivalent 1 a is the of extended real numbers a which are the limits of subse quences of an 39 2 For every 6 gt 0 a there is a N such thatfor all natural n 2 N an gt aie and b for an in nite number of indices n an lt a e 3 a sulp ak If any and hence all of properties 173 hold a is called the lower limit or limit inferior of a sequence an a anl Dually the following theorem holds Theorem 2 Let an be a sequence of real numbers anda E R Then the following statements are equivalent 1 a is the supremum of extended real numbers a which are the limits of sub sequences of an 2 For every 6 gt 0 a there is a N such that for all natural n 2 N an lt ae and b for an in nite number of indices n an gt a 7 e 3 a inf sup ak n IcZn If any and hence all of properties 173 hold a is called the upper limit or limit superior of a sequence an a m a Upper and lower limits have the following properties 1 mean imam 2 inf an S man S liman S sup an 3 man Man if and only if lim an exists In this case man Man lim anl 4 Man li7mbn S Emu bn S Man mbm provided the right and left sides are not of the form 00 7 00 5 if an 2 0 and b7 2 0 then Manhimbn S mmnbn S Manmbn provided the product on the right is not of the form 0 00 Problem 1 Find m71 ni1 and lifm71 ni1


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