Let S be a bounded, nonempty set of real numbers. Prove each of the following: (

Question

Let S be a bounded, nonempty set of real numbers. Prove each of the following: (a) There exists a sequence of points xn ? S such that lim xn = sup S. n?? (b) If {xn} is any sequence in S that converges, then inf S ? lim xn ? sup S.

Explanation / Answer

folow this Proof: S is a nonempty set of real numbers that is bounded from above and X = sup S. Thus by definition of supremum, we have, that if x = supS, there exists an s(?)S such that for every ? > 0, x-? s(?). If xS (this is possible because the supremum might lie in the same set), then we are done. Otherwise, x does not belong to S. Thus we get a sequence of x such that x-? s(?). Thus we have infinitely many values of S in the ?-neighbourhood of x. Hence, by definition of accumalation point, x is an accumalation point. Report Abuse

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.