Question regarding the formal definition of limes Inferior/Superior

Question

I have a question regarding the definitions of limes inferior/superior of a sequence $x_n$.

Limes inferior for example is defined as

$$lim ~ sup_{n \rightarrow \infty} x_n := lim_{n \rightarrow \infty} (sup\{x_k |k \ge n\})\in \mathbb{R} \cup \{-\infty\}$$

Now I simply do not understand why the condition $k \ge n$ is needed. Is it meant as the new sequence created by the limes inferior kind of goes ahead of the actual sequence $x_n$ and checks its values in advance?

It really confuses me at the moment.

Thank you very much for your help.
FunkyPeanut

Answer 1

I give you here an intuitive description as it seems it is what you asked for (I swapped the $n$'s and the $k$'s sorry).

Reason: one of the main ideas is that you want to define a limit which exists (which might be infinity), by defining a $\limsup$ of a sequence you pick the highest point of your sequence as it goes to infiny and therefore you pick a point of your sequence, no mater if you limit exists or not.

An intuitive way to compute is to split the process in two steps:

1. take the collection of all subsequences of the form $X_k=\{x_n\}_{n\geq k}$ $k=1,2,...$. The sup exists for all $X_k$ (note that $\sup_n X_k$ is monotonically decreasing in $k$, and so taking the limit cannot give rise to problems related to oscilating sequences for example).
2. Now you want to pick the last supremum of your sequence (the highest point of your sequence as it goes to infinity) ans so take the limit of $X_k$ in $k$.

The easiest example to see how this can be useful is to take the sequence $\{1,-1,1,-1,1,...\}$.

In your comment you are right to say that a new sequence is created it is the sequence $\{\sup_n\{x_n\}_{n\geq k}\}_{k\geq 1}$