Theorem 4.5.6. Let (S, d) be a metric space with A C S. Then x € A if and only if there exists asequence (xn :n E N) in A that converges to x.Let's convince ourselves that this Theorem is true.Since it is an if and only if statement, the Theorem says two things:1. if æ € Ā then there exists a sequence (xn :n E N) in A that converges to æ.2. if there exists a sequence (x, : n E N) in A that converges to x, then æ € A.Explain why any ball B(x, 1/n) will have a nonempty intersection with A in the discussion above.We mentioned in the discussion above that there exists a natural number N such that 1/N< e, for any e> 0. Convince yourself of thisfact. If e= 0.01, what value could you choose for N so that 1/N< e?In the discussion of part 1) above, why is it essential that Xn eB(x, e) NA and not simply n eB(x, e)?

Name: Theorem 4.5.6. Let (S, d) be a metric space with A C S. Then x € A if
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Description: Theorem 4.5.6. Let (S, d) be a metric space with A C S. Then x € A if and only if there exists asequence (xn :n E N) in A that converges to x.Let's convince ourselves that this Theorem is true.Since it is an if and only if statement, the Theorem say

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Theorem 4.5.6. Let (S, d) be a metric space with A C S. Then æ € A if and only if there exists a sequence (xn : n e N) in A that converges to æ. Let's convince ourselves that this Theorem is true. Since it is an if and only if statement, the Theorem says two things: 1. if æ e A then there exists a sequence (æn :n € N) in A that converges to aæ. 2. if there exists a sequence (x, :n E N) in A that converges to aæ, then a E A. Explain why any ball B(x, 1/n) will have a nonempty intersection with A in the discussion above. We mentioned in the discussion above that there exists a natural number N such that 1/N < «, for any e>0. Convince yourself of this fact. If e= 0,01, what value could you choose for N so that 1/N < e? In the discussion of part 1) above, why is it essential that Xn eB(x, e) NA and not simply Xn eB(x, e)?

Theorem 4.5.6. Let (S, d) be a metric space with A C S. Then æ € A if and only if there exists a sequence (xn : n e N) in A that converges to æ. Let's convince ourselves that this Theorem is true. Since it is an if and only if statement, the Theorem says two things: 1. if æ e A then there exists a sequence (æn :n € N) in A that converges to aæ. 2. if there exists a sequence (x, :n E N) in A that converges to aæ, then a E A. Explain why any ball B(x, 1/n) will have a nonempty intersection with A in the discussion above. We mentioned in the discussion above that there exists a natural number N such that 1/N < «, for any e>0. Convince yourself of this fact. If e= 0,01, what value could you choose for N so that 1/N < e? In the discussion of part 1) above, why is it essential that Xn eB(x, e) NA and not simply Xn eB(x, e)?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Theorem 4.5.6. Let (S, d) be a metric space with A C S. Then x € A if and only if there exists a
sequence (xn :n E N) in A that converges to x.
Let's convince ourselves that this Theorem is true.
Since it is an if and only if statement, the Theorem says two things:
1. if æ € Ā then there exists a sequence (xn :n E N) in A that converges to æ.
2. if there exists a sequence (x, : n E N) in A that converges to x, then æ € A.
Explain why any ball B(x, 1/n) will have a nonempty intersection with A in the discussion above.
We mentioned in the discussion above that there exists a natural number N such that 1/N< e, for any e> 0. Convince yourself of this
fact. If e= 0.01, what value could you choose for N so that 1/N< e?
In the discussion of part 1) above, why is it essential that Xn eB(x, e) NA and not simply n eB(x, e)?

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