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

We are given the following theorem. Let (S, d) be a metric space with A⊆S. Then x∈A if and only if…

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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

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A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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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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3.5.6
2. Consider the function y(x) = 5x over the interval [0, 10]. (a) If the points A(0, y(0)), B(5, y(5)), and C(10, y(10)) are on the curve represented by the above function, find the slope of the lines (secants) AB, and AC, respectively. (b) If x represents time and y represents the displacement of a particle moving along a line, what is the physical meaning of your results to part (a)? (c) Now consider a point D(10 – h, y(10 – h)) on the curve represented by the given function, find the slope of the line DC. (d) As h → 0, what is the limiting value of your result to part (c)? (e) What is its physical meaning of your result to part (d) if again x represents time and y represents the displacement?
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