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6. Give a two-part two-part direct proof outline of the following claim, based only on the logical structure of the claim and on the given definitions. CLAIM. Let SCR. A number a is an accumulation point for S if and only if for any 8 > 0, N(x, 8) contains infinitely many elements of S. Definitions: • For any a E R and & > 0, N(x, 8) := {y €R | |a – y| < 8}. • A number r is an accumulation point for a set S iff for any 8 > 0, N(r, 8) contains at least one element of S which is distinct from r. For each of the remaining items, give a complete proof of the specified form. You need not include your proof outline. Use algebra for your arguments; there is no need to specify any axiom of real numbers used.

6. Give a two-part two-part direct proof outline of the following claim, based only on the logical structure of the claim and on the given definitions. CLAIM. Let SCR. A number a is an accumulation point for S if and only if for any 8 > 0, N(x, 8) contains infinitely many elements of S. Definitions: • For any a E R and & > 0, N(x, 8) := {y €R | |a – y| < 8}. • A number r is an accumulation point for a set S iff for any 8 > 0, N(r, 8) contains at least one element of S which is distinct from r. For each of the remaining items, give a complete proof of the specified form. You need not include your proof outline. Use algebra for your arguments; there is no need to specify any axiom of real numbers used.

Advanced Engineering Mathematics
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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6. Give a two-part two-part direct proof outline of the following claim, based only on the logical structure of the claim and on the given definitions. CLAIM. Let SCR. A number a is an accumulation point for S if and only if for any 8 > 0, N(x, 8) contains infinitely many elements of S. Definitions: • For any a ER and &> 0, N(x, 8) := {y €R | |r – yl < 8}. • A number r is an accumulation point for a set S iff for any 8 > 0, N(x, 8) contains at least one element of S which is distinct from r. For each of the remaining items, give a complete proof of the specified form. You need not include your proof outline. Use algebra for your arguments; there is no need to specify any axiom of real numbers used.
Transcribed Image Text:6. Give a two-part two-part direct proof outline of the following claim, based only on the logical structure of the claim and on the given definitions. CLAIM. Let SCR. A number a is an accumulation point for S if and only if for any 8 > 0, N(x, 8) contains infinitely many elements of S. Definitions: • For any a ER and &> 0, N(x, 8) := {y €R | |r – yl < 8}. • A number r is an accumulation point for a set S iff for any 8 > 0, N(x, 8) contains at least one element of S which is distinct from r. For each of the remaining items, give a complete proof of the specified form. You need not include your proof outline. Use algebra for your arguments; there is no need to specify any axiom of real numbers used.
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