I am taking an introductory course in Real Analysis and
$(i)$ My textbook gives the following definition of a limit superior $U$ which satisfies the following conditions:
$(a)$ For every $\epsilon>0$,there exists an integer $N$ such that $n>N \implies a_n<U+\epsilon$
$(b)$ Given $\epsilon>0$ and given $m>0$, there exists an integer $n>m$ such that $a_n>U-\epsilon$
$(ii)$ An another textbook gives the following definition of a limit superior of a sequence :
The highest limit point of a given sequence is called the limit superior of the sequence.
$(iii)$ I read somewhere else that :
limit superior is the infimum of those real numbers $v$ with the property that there are a finite number of natural numbers $n$ such that $v <x_n$.
EDIT : Attempt: Suppose $x_n>v$ for infinite terms of the sequence $X=\{x_n\}$ and $v$ is the highest limit point of the sequence.
I was thinking of making use of Bolzanno Weirstrass Theorem, but we would need $X$ to be bounded for that. How do I bring a contradiction?
I am confused about these definitions and not able to make any connection between definitions $(i),(ii),(iii)$.
Why is it the highest limit point and contain only a finite number of points of the sequence to the right?
How do I prove that these all are equivalent?
Thank you for your help..