What is the symbol ε in this definition of a limit superior?
I tried to understand it through given statemets about limit superior such as in $(i)$ where if all members of the sequence satisfy $$x_n < L+ε,$$ then perhaps ε should be the difference between the supremum of this sequence and its limit superior.
But then according to $(ii)$ $$x_n > L-ε,$$ ε should also be the difference between this sequence's limit superior and its infimum?
Which perhaps is not contradicting the statement in $(i)$ but to me it sounds as if it does for some reason.
It would be of great help if someone could define this ε.
Thank you for your time and help in advance.
I think an example will serve to clarify:
Take the sequence to be $\{x_n=\frac{1}{n}\}_{n=1}^\infty$
The limsup of the sequence is 0.
why?
$(\forall \epsilon>0)$ only a finite number of terms satisfies $x_n>0+\epsilon=\epsilon\implies x_n>\epsilon$.
namely, $n<\frac{1}{\epsilon}$.
This is condition i. of the definition.
Also an infinite number of terms satisfies:
$x_n>-\epsilon$.
Infact all terms are non-negative.
This is condition ii. of the definition.