6 Exercise 6: | Let S be a monempty subset of 1R that is boundedbelow • Pove that if S = - sup { - 5:5€5}.

Let be a non-empty subset of that is bounded below.Then, Infimum of S exists and suppose…

Answered: Exercise 6: Let S be a monempty subset…

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

Exercise 6: | Let S be a monempty subset of 1R that is bounded
below • Pove that if S = - sup { - 5:5€5}.

Expert Solution

Step 1: ''Introduction to the solution''

Let $S$ be a non-empty subset of $straight real numbers$ that is bounded below.

Then, Infimum of S exists and suppose Inf $open parentheses S close parentheses$ $equals m$ .

So, by the definition of Infimum , we obtain (1) $m less or equal than s comma space for all s element of S.......... left parenthesis 1 right parenthesis$

and $open parentheses 2 close parentheses$ $for all epsilon greater than 0 comma space$ there exists an element $y element of S space$ such that $y less than left parenthesis m plus epsilon right parenthesis............ left parenthesis 2 right parenthesis$

To show $I n f left parenthesis S right parenthesis equals negative S u p left curly bracket negative s colon space s element of S right curly bracket$ , it is enough to show that $S u p left parenthesis negative S right parenthesis equals negative I n f left parenthesis S right parenthesis equals negative m$ .

where $S u p left parenthesis negative S right parenthesis equals left curly bracket negative s colon space s element of S right curly bracket$