Answered: Show that any nonempty set of integers…

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

Show that any nonempty set of integers that is bounded above has a largest member.

Expert Solution

Step 1: To Apply the Well-Ordering Principle:

$L e t space A space i s space n o n minus e m p t y space s e t space o f space i n t e g e r s. A space i s space b o u n d e d space a b o v e. H e n c e comma space there exists space M element of straight integer numbers comma space s o space t h a t space for all a element of A comma space a less or equal than M. W h i c h space c l e a r l y space g i v e s space 0 less or equal than M minus a comma space for all a element of straight integer numbers. L e t comma space B equals left curly bracket M minus a colon a element of A right curly bracket. T h e r e f o r e space B subset of or equal to straight natural numbers. B y space W e l l minus O r d e r i n g space P r i n c i p l e comma space B space h a s space a space l e a s t space e l e m e t. l e t space b space i s space t h e space l e a s t space e l e m e n t space o f space B.$