Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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i+2+3+ ·+n = zn(n+1) 5. Let S be a nonempty subset of R that is bounded above, with upper bound b. Prove that b = sup(S) if the following condition holds : for every e > 0 there is IE S such that b – e < x < b. To prove this, assume that there is some real number c with the property that c2 x for every r E S and that c < b. Show that this assumption leads to a contradiction, and explain why the contradicted assumption proves that b= sup(S).
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Transcribed Image Text:i+2+3+ ·+n = zn(n+1) 5. Let S be a nonempty subset of R that is bounded above, with upper bound b. Prove that b = sup(S) if the following condition holds : for every e > 0 there is IE S such that b – e < x < b. To prove this, assume that there is some real number c with the property that c2 x for every r E S and that c < b. Show that this assumption leads to a contradiction, and explain why the contradicted assumption proves that b= sup(S).
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