Question 1.3.6. Introduction to analysis by W. Wade. 1. Prove that if a set E wh

Question

Question 1.3.6. Introduction to analysis by W. Wade.
1. Prove that if a set E which belongs to R has a finite infimum and E>o is any positive number, then there is a point a belonging to E such that infE+E>a>=inf E.
Use approximation property for infima
2. if E is a subset of R and is nonempty and bounded below, prove that E has a finite infimum.
Use completeness property for infima

Explanation / Answer

1) If inf E is in E, we are done since we can set a=inf E. So, say inf E is not in E (the set, not the number E). To avoid confusion, re-label the positive number as V. Given any w > 0, there exists a number a in E such that inf E inf E. 2) Well, usually the completeness axiom states that any set bounded above has a least upper bound (sup.) It is an axiom, not something one proves. If E is a set bounded below, then let A = { -x : x in E} . A a set bounded above. A has a least upper bound by completeness, say V = sup A. The one can show that -V is the inf of E.

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