6. Suppose that f:R → R is uniformly continuous so that for some o > 0, and for all r, y ERI/(1) – f(u)| <1 whenever |r – y| < 8.If f(0) = 0, Prove that for every a > 0f(2)| <1+ r/8.Hint: For each r> 0 show that |f(r)| <1+ng, where n, is the greatest integer not exceeding a/6.7. Define the sequence (a,) as follows: For every natural number n, leta, = [1.7+ (-1)"]".Does lim sup a, exist? What about lim inf a,? Find the set of subsequential limits for the sequence (a,).

The objective here is to prove: if f(0)=0then for every x>0 fx<1+xδ Given: f:R→R is uniformly…

Answered: 6. Suppose that f: R -R is uniformly…

6. Suppose that f: R -R is uniformly continuous so that for some 6 > 0, and for all 1, y € R I/(=) – f(y)| <1 whenever |r – y| < 6. If f(0) = 0, Prove that for every z>0

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 30EQ

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Question

6. Suppose that f:R → R is uniformly continuous so that for some o > 0, and for all r, y ER
I/(1) – f(u)| <1 whenever |r – y| < 8.
If f(0) = 0, Prove that for every a > 0
f(2)| <1+ r/8.
Hint: For each r> 0 show that |f(r)| <1+ng, where n, is the greatest integer not exceeding a/6.
7. Define the sequence (a,) as follows: For every natural number n, let
a, = [1.7+ (-1)"]".
Does lim sup a, exist? What about lim inf a,? Find the set of subsequential limits for the sequence (a,).

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