{ne N} is neither open nor closed; 1 Let S = {:n € N} and take xo € S. To show that S is not open, prove that, for all € > 0, D (xo, €) S- Explain why? To show that for all e > 0, D (xo, e) S, you may consider the following steps: . Observe that co= 1, for some N € N. • Notice that D (x0, €) = ( − €, ½ + €) - Explain why? • Use the fact that an open interval in R is not countable to show that D (xo, e) S. What is the cardinality of S'? To show that S is not closed, prove that S = R\S is not open. To show that Se is not open, prove that, for all € > 0, D (0, €) Sc - Why is this sufficient? Please explain. • Notice that D (0, €) Sc iff D (0, €) • Observe that 0 € Sc and take € > 0. there is NE N, such that 0 << €. • Show that there is x E D (0, e) S. What can you take for x? SØ - Explain why? Which property of real numbers will you use to show that
{ne N} is neither open nor closed; 1 Let S = {:n € N} and take xo € S. To show that S is not open, prove that, for all € > 0, D (xo, €) S- Explain why? To show that for all e > 0, D (xo, e) S, you may consider the following steps: . Observe that co= 1, for some N € N. • Notice that D (x0, €) = ( − €, ½ + €) - Explain why? • Use the fact that an open interval in R is not countable to show that D (xo, e) S. What is the cardinality of S'? To show that S is not closed, prove that S = R\S is not open. To show that Se is not open, prove that, for all € > 0, D (0, €) Sc - Why is this sufficient? Please explain. • Notice that D (0, €) Sc iff D (0, €) • Observe that 0 € Sc and take € > 0. there is NE N, such that 0 << €. • Show that there is x E D (0, e) S. What can you take for x? SØ - Explain why? Which property of real numbers will you use to show that
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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Please follow all the steps and include all necessary explanations for better understanding. Thanks
Transcribed Image Text:{ n =N} is neither open nor closed;
1
Let S = {:n € N} and take xo € S. To show that S is not open, prove that, for all € > 0,
D (xo, €) S- Explain why?
To show that for all e > 0, D (xo, e) S, you may consider the following steps:
. Observe that co= 1, for some N € N.
• Notice that D (x0, €) = ( − €, ½ + €) - Explain why?
• Use the fact that an open interval in R is not countable to show that D (xo, e) S. What is the
cardinality of S'?
To show that S is not closed, prove that S = R\S is not open. To show that Se is not open, prove
that, for all € > 0, D (0, €) Sc - Why is this sufficient? Please explain.
• Notice that D (0, €)
Sc iff D (0, €)
• Observe that 0 € Sc and take € > 0.
there is NE N, such that 0 << €.
• Show that there is x E D (0, e) S. What can you take for x?
SØ - Explain why?
Which property of real numbers will you use to show that
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