6. For n E N, define 9n : R → R byIn(x) ==n1,9if |x| ≤ nif |x| > n(a) Show that the sequence {n}_₁ is uniformly bounded.(b) Show that the sequence {n}_₁ is equicontinuous.n=1(c) Show that the sequence {n} does not have a convergent subsequence.∞n=1

To show that the sequence {gn} is uniformly bounded, we need to find a constant M such that for all…

Answered: 6. For n E N, define gn R R by -{ :=…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 64E

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6. For n E N, define gn R R by -{ := In(x): |x| n 1, if x ≤ n if |x| > n (a) Show that the sequence {n}_₁ is uniformly bounded. (b) Show that the sequence {n}a_1 is equicontinuous. (c) Show that the sequence {gn}=1 does not have a convergent subsequence.