Consider the set U = {(x, sin(1/x)) : 0 < x} C R². a) Show that every point on the vertical line segment given by {(0, y) : –1
Consider the set U = {(x, sin(1/x)) : 0 < x} C R². a) Show that every point on the vertical line segment given by {(0, y) : –1
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 21E
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![Consider the set U = {(x, sin(1/x)) : 0 < x} C R².
a) Show that every point on the vertical line segment given by {(0,y) : –1 <y< 1} is a limit point of U.
Hint: Show that for any -1 < y< 1, there exists a sequence (xn) that converges to 0 with sin(1/xn) = y for
each n.
b) Is U closed?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F096e3465-5e66-41e8-8526-c018fcf73d9b%2F0662756e-733e-4f4b-8caf-b2b62acaad22%2F0btgkkt_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the set U = {(x, sin(1/x)) : 0 < x} C R².
a) Show that every point on the vertical line segment given by {(0,y) : –1 <y< 1} is a limit point of U.
Hint: Show that for any -1 < y< 1, there exists a sequence (xn) that converges to 0 with sin(1/xn) = y for
each n.
b) Is U closed?
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