# Proof that the limit superior of a function is a point of accumulation

In my complex analysis book, the author asks the reader to prove that the limit superior of a series is a point of accumulation. I know how to prove that it is the largest such point but I can't figure out how to prove that it is one in the first place.

We are not suppose to use the fact that for a given series Tn with a point of accumulation t it holds that there are infinitely many $$n$$ such that $$T_n\ge t - e$$ and finitely many n such that $$T_n\ge t + e.$$

My attempt:

I can justify it in my own head since I know that a second definition for a point of accumulation is a limit point and since the limit superior is a upper limit point it should therefore be an point of accumulation as well. But I don't know how to prove that mathematically.

My definitions:

I wanted to add this section because I feel like I could be misunderstanding some of the definitions:

Limit superior - the least upper point ( I am picturing a sequence that oscillates and the limit superior being the up-most point of the limit as n tends to infinity of that sequence)

Point of accumulation - A point such that there exist infinitely many indiecies such that $$|t_n -t| i.e., from the formal definition of the limit, it is an adherent limit point of that sequence.

Complications :

Even assuming that the limit superior of a function is a point of accumulation I see some contradictions. For example it states that this point t of a sequence tn holds

there exist finitely many indices such that $$t_n\ge t -e$$

=> there exist infinitely indices such that $$t_n\le t -e$$

=> $$t_n -t \le -e$$

Now to omit the definition of a point of accumulation I will have to take the modulus of both sides:

$$|t_n - t|\le |e|=e$$ but I don't think that is a valid step because for example $$-3<-2$$ but $$3<2$$ is false. So in order for it to hold this property, I will have to perform an "illegal" operation. I am sorry if this is too confusing if anybody could help me I would greatly appreciate it.

• It'd help to start with a precise definition of the limit superior. There are a few different ones, which does your text/course use? – Math1000 May 13 '19 at 20:43
• @Math1000 "Let S be the set of accumulation points. We define the limit superior of the sequence to be the least upper bound of S." – Maths Wizzard May 13 '19 at 20:49