12:21 PM ..ooo Sprint LTE Back 3318 LPSnotes.pdf Chapter 5: Limit Points A point

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12:21 PM ..ooo Sprint LTE Back 3318 LPSnotes.pdf Chapter 5: Limit Points A point p is said to be a limit point of a point set M if and only if every region noutaining contains a point of M distinct from p. The set of all limit points of a point set M is denoted Complete: The statement point pisnotalimit point of apoiat M meaak. Question 5.1 Exercise 5.1. Read the definition carefully (a) Is it specified whether or not PE M? (l) Does the regiou that is mentioned have to be a sulouet of Al? Exercise 5.2. Let A 10, 10l. B (0,10), C (0,10 D 1,213 9, 10) a. In S-R, is 10 a limit point of A? B? C? D? Explain. b. IaS R, is 3 a limit point of A? D? (0,3 U(3,10? Explain. c. S z, is 10 a limit point of D? of 3,5,79)? Explain. d. laS z, is 3alinait point of D2, of (1,3,5, 7,9)? Explain. Problem 5.1. Prove that pe in each example below, where S R. 05 and 0 and Problem 5.2. Prove that pe MI', where S R. p o and M 1/2, 1/3,...). Problem 5.3. Suppose S AUB, where A e RIz lews than 0) and B a ER is greater than or equal to 1). By precede" we mean the natural order of real numbers. the Prove that 1 E A'. 5.1. If the point set H is a subset of the point set K. then H CK Theorem 5.2. If H and K are point sets and Pe (HuK). then PE H orPEK Theorem 5.3. If pis a limit point of the union of a empty, faite collection 9 of point sets, then p is a limit point of at least one member of Theorem 5.4. No finite point Met has a limit point. Problem 5.4. Using the real numbers as the model, which of the following are true and which are not true? a. Some finite point set has a limit point. b. Any infinite number set M has a limit point. Any suhteet of la has a limit point. d. All limit points of a given set bekeng to the set. Theorem 5.5. If H is a point set, K is a finite point set, and PE (HUKY, then PEH. Theorem 5.6. If p is a limit point of the point set M. the each region coetaining p coetains intaitely many different points of M Courses Calendar To Do Notifications Messages

Explanation / Answer

The definition of limit point is given. so, we just have to negate the definition.

Negation is as follows:

A point p is not a limit point of a set M if there is some region containing p whose every element q, different from p lies outside the set M.

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