Basically, I am having trouble understanding this hole page from my complex analysis book. I fail to understand why a point of accumulation of a given set is also its limit supperior and most importantly why it holds the properties mentioned on the page and I am also not understanding the proof provided by the writer.
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.
Here is the page: Extract from the book