inf a, + inf b,

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 15E
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How do you solve the second statement/bullet point?

For a sequence (an), we define
sup an
= sup{an,n > 1}, infa, := inf{a,,n > 1}.
• Show that
sup(-an) = - inf an, inf(-an) = – sup an.
n
Show that
inf a, + inf b, < inf(a, + bn), sup(an + bn) < sup an + sup bn,
n
n
n
and give an example where the first inequality is strict. (Hint: using the first part, you only need to show one of
them.)
Transcribed Image Text:For a sequence (an), we define sup an = sup{an,n > 1}, infa, := inf{a,,n > 1}. • Show that sup(-an) = - inf an, inf(-an) = – sup an. n Show that inf a, + inf b, < inf(a, + bn), sup(an + bn) < sup an + sup bn, n n n and give an example where the first inequality is strict. (Hint: using the first part, you only need to show one of them.)
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