Let K be compact in R and let c R. Prove that there exists

Question:

Let K ≠ θ be compact in R and let c ∈ R. Prove that there exists a point b in K such that |c - b| = sup{|c - x| : x ∈ K}.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Introduction to Real Analysis

ISBN: 978-0471433316

4th edition

Authors: Robert G. Bartle, Donald R. Sherbert

See More Books

Question Posted: Apr 07, 2016 11:13 AM

See More Questions

Let f be continuous on the interval [0, 1] to R and such that f(0) = f(1). Prove that there exists a point c in [0, 1/2] such that f(c + 1/2). [Consider g(x) = f(x) - f(x + 1/2).] Conclude that there...
Let E be closed and bounded in R, and suppose that for each x E there is a function fx, nonnegative, nonconstant, increasing, and C on R, such that fx(x) > 0 and fx(y) = 0 for y E. Prove that there...
Let I := [a, b and let f : I R be a continuous function such that f(x) > 0 for each x in I. Prove that there exists a number a > 0 such that f(x) > a for all x I.
Which of these situations would require auditors to append an emphasis- of- matter paragraph about consistency to an otherwise unmodified opinion? a. Entity changed its estimated allowance for...
Roy and Icie Johnson established two revocable inter vivos trusts in 1966. The trusts provided that upon Roy's and Icie's deaths, income from the trusts was to be paid in equal shares to their two...
Which polar coordinate pairs label the same point? a. (3, 0) b. (-3, 0) c. (2, 2/3) d. (2, 7/3) e. (-3, ) f. (2, /3) g. (-3, 2) h. (-2, -/3)
Identify factors that influence financial statements and their analysis. (p. 428) LO.1
Exeter, Inc., is a building contractor on the Gulf Coast. After losing a number of big lawsuits, it was facing its first annual net loss as the end of the year approached. The CEO, Hank Snow, was...
Paragraph about The Best Kind of Weather . Include at least one compound sentence and one complex sentence in paragraph. 500 words
The following balance sheets have been prepared as at December 31, Year 6, for Kay Corp. and Adams Ventures: Additional Information: Kay acquired its 40% interest in Adams for $374,000 in Year 2,...
Let K , be a compact set in R. Show that inf K and sup K exist and belong to K.
Use the notion of compactness to give an alternative proof of Exercise 5.3.18.
In the past, Taylor Industries has used a fixedtime period inventory system that involved taking a complete inventory count of all items each month. However, increasing labor costs are forcing Taylor...
?In civil engineering, what is the main use of a slump test in concrete technology?
Briefly explain what spatial autocorrelation means and what method can be used to measure it
Discuss the theoretical implications of adopting biodegradable materials in civil engineering for reducing environmental impact and enhancing sustainability.
Analyze the role of civil engineering in coastal erosion management. What are the engineering strategies for shoreline protection, beach nourishment, and coastal infrastructure design to mitigate...
Analyze the impact of climate change on civil engineering practices, particularly in the areas of coastal and floodplain management, and discuss strategies for mitigating these impacts
What do you mean by callback? How will you implement callback mechanism in CORBA?
Consider a game of poker being played with a standard 52-card deck (four suits, each of which has 13 different denominations of cards). At a certain point in the game, six cards have been exposed. Of...
List in order from least to the greatest. -8 7/8, 7 (to the 1st power), -5, |-6|, 4, |3|, - 8 5/8, - 100, 0, 1 (to the 7th power), 14/4, -67/8
-17.8 / 3.2
-5/6 + (-2/9)
Choose two stocks from the same industry to minimize the influence of other confounding factors. You choose the industry that you are relatively more familiar with, and then estimate the implied...
why should Undertake research to review reasons for previous profit or loss?
A pension fund's liabilities has a PV01 of $200 million. The plan has $100 billion of assets with a weighted average modified duration of 8. The highest duration bond that the plan can invest in has...

Let K be compact in R and let c R. Prove that there exists

Question:

Step by Step Answer:

Let K R be compact and let c R If n N there exists x n K such that sup...View the full answer

Introduction to Real Analysis

Students also viewed these Calculus questions