Prove the following theorem/statements (1) If x + 0, then +0 and TV = x.(2) Let a, b e R. a² + b² = 0 if and only if a = 0 and b = 0.(3) Let x, y E R and suppose for every ɛ > 0, we have x 0 for all x € F. Conclude from this that is x? +1 = 0 has asolution in F, then the field F cannot be ordered.(6) The Archimedean Property states that the set N of natural numbers is unbounded above in R.It is equivalent to the following. (*)(a) For each z E R, there exists an n E N such that n > z.(b) For each x > 0 and for each y E R, there exists an n E N such that nx > y.(c) For each > 0, there exists an n E N such that 0 < < x.

Note: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts…

Answered: (1) If x + 0, then +0 and A = x. (2)…

(1) If x + 0, then +0 and A = x. (2) Let a, b e R. a² + b² = 0 if and only if a = 0 and b = 0. %3D (3) Let x, y ER and suppose for every e > 0, we have x < y+ɛ. Then r < y.(*)

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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Question

Prove the following theorem/statements

(1) If x + 0, then +0 and TV = x.
(2) Let a, b e R. a² + b² = 0 if and only if a = 0 and b = 0.
(3) Let x, y E R and suppose for every ɛ > 0, we have x <y+ ɛ. Then x < y.(*)
(4) If |x – y| < c, then |x| < |y| + c.
(5) In any ordered field F, x2+1> 0 for all x € F. Conclude from this that is x? +1 = 0 has a
solution in F, then the field F cannot be ordered.
(6) The Archimedean Property states that the set N of natural numbers is unbounded above in R.
It is equivalent to the following. (*)
(a) For each z E R, there exists an n E N such that n > z.
(b) For each x > 0 and for each y E R, there exists an n E N such that nx > y.
(c) For each > 0, there exists an n E N such that 0 < < x.

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