Limit point Definition 24 Suppose that K is a point collecti

Limit point Definition 24. Suppose that K is a point collection. The statement that L is a limit point of K means L is a point and if (S, T) is a plane segment that contains L, then there is a point in K, distinct from L, in PROBLEM 19. Suppose M is an infinite subset of the plane interval Ila, b] [c, d], then M has a limit point. Explain your answer using English sentences Sequencial limit point Definition 25. Suppose that A is an point sequence. The statement that L is a sequencial limit point of A means: (1.) L is a point; and ( 2.) if (S, T) is a plane segment containing L, then there is a number N, such that if n is a positive integer greater than N, then An is in (S,T). PROBLEM 20. Suppose M is subset of the plane segnent PS and M has a limit point Q. Then there is a point sequence A, each term of which is in M, such that is a sequential limit point of A. Explain your answer using English sentences.

Solution

Question 20.

M is a subset of the plane segment PS and M has a limit point Q

Therefore, Q is a point and the plane segment PS contains Q.

Say, A is a point sequence in M, i.e. A is a subset of M

Therefore, A is a subset of the plane segment PS.

Hence, there exists natural number N such that A_n is in PS for all n > N

Therefore by definition 25, Q is a sequencial limit point of A. (Proved)