Definition of the Limes Superior

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I understand what the limes superior is insofar as I know that for example for an alternating series as $a_n = (-1)^n$ which does not diverge to $\infty$, we can define the bigger accumulation point 1 as the limes superior.

What I don't understand yet though is the corresponding notation which is:

$\sup_{n \in \mathbb N} \ \inf_{k \geq n} x_k$

I understand this so far as follows:

1. I pick one concrete $n \in \mathbb N$, say n = 3.
2. Then I pick the infimum of the set $\{x_3,x_4,.....x_{1000},...\}$, lets call this infimum $\inf_3$
3. Next I set n = 4. And pick the infimum of the set $\{x_4,x_5,....\}$ etc.

Naturally this sequence of infima must be monotonically falling (each set of we take the infimum of is a subset of the previous set). So, of what exactly are we then taking the Supremum of? I.e. where is my understanding wrong, since such as I understand it obviously is not making much sense...thanks

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1 Answer

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The sequence is monotonically not decreasing, actually. Think at the sequence $(2 - 1/n)$; for $n=1$ the inf is 1, for $n=2$ is 3/2, and so on.

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