Question

Let{an}[infinity]n=1be a sequence. A real numberxis alimit point(sometimes called anaccumulation point) if there is a subsequence{ank}[infinity]k=1which converges tox.(a) How many limit points do the following sequences have?(i){(−1)n}[infinity]n=1(ii)an= 10 ifn= 1, . . . ,100 andan=1nifn >100.(b) Construct a sequence that does not have a limit point.(c) Construct a sequence with exactly 2 limit points.(d) Construct a sequence that has exactly one limit point, but which does not con-verge.

Answer 1

Answer:

Answer for the question is explained din the attachment.

Step-by-step explanation:

a. i. Sequence has only two limit point

ii. Sequence is convergent and have only one limit point.

b. We have maked sequence that not have a limit point.

c. We have maked sequence that has two limit point.

d. We constructed a sequence that has only one limit point but not convergent.