I hope this is the right forum, I wasn't sure exactly where to post this. The course is Analysis I. I wrote a proof for this problem, but I am new at these and I'm unsure whether my logic holds. All we are allowed to use for this assignment is the definition of limit point. Here is what I have.
Problem: Show that if is the set of all positive integers, then no point is a limit point of .
Solution: Assume that a limit point of . Then all open intervals containing must contain an element of different from . Consider the interval where and and . That means a point such that . But if , then no such exists because any point between and is not an integer. Thus, not every open interval containing contains a point of different from , therefore no limit point of exists.
My question is mainly in regards to the case where I suggest because does that also mean I'm implying that which isn't necessarily true? Or is this okay?