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!)
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.