Dan, are you quite sure that you have quoted this problem correctly?

It clear that

can be a limit point of the set

without being the limit of the sequence

. Unless you textbook is making some unusual point, I don’t think that statement is true. (Or maybe I have misread the statement!)

**Post Script **
I have just thought of a reading of the phrase “given any

the inequality

holds for infinitely many points of

”. If we read that as meaning

*infinitely many *__terms__ of the sequence then any sequence that is constant from some index on will fit that reading. Here is a trivial example:

. The number

**i** is limit of the sequence but not a limit point.