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Asked Aug 10, 2019
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I need help for number 4. Thanks!

(a) Prove that |x | a ll. (Hint: See Exercise 1.2.17.)
(b) Prove that if b e R" is any vector, then b xk b a
#4. Prove thata rectangle R = [a1, bi] x ...x [an, bn] C R" is closed.
*5. Prove that the closed ball B (a, r) = {x e R" ||x - a|| < r} C R" is closed.
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(a) Prove that |x | a ll. (Hint: See Exercise 1.2.17.) (b) Prove that if b e R" is any vector, then b xk b a #4. Prove thata rectangle R = [a1, bi] x ...x [an, bn] C R" is closed. *5. Prove that the closed ball B (a, r) = {x e R" ||x - a|| < r} C R" is closed.

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Expert Answer

Step 1

We need to show that

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x [an, bn c R" is closed. [a1, b] x a rectangle R Note: A set A C R" is closed if and only if every convergent sequence in R" completely contained in A has its limit in A

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Step 2

Now,

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Let {xn CR be a convergent sequence in R. Let us assume that x. So, by the above statement if we can show that x ER then we are done Now VE 0,3 No > 0 such that |2n - xl <e for every n> No Хn _ - (х,, хв xand x denotes the i-th co-ordinate. 2 (x1,x2, .., x") where ...., xn") and x = Let us denote xn i

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Step 3

Further,...

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1/2 <E (by the Euclidean norm ) => Zi=1 nx e2 (by squaring both sides) I* - xl < /vm => =>

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