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How do we construct the limsup and the liminf of a sequence of functions? I don't understand why should we consider a sequence of infima or suprema of the functions whose indices are greater or equal than a given number. My book says that this number is exactly the supremum or the infimum of the sequence. As an example, the liminf is defined as:

$$\liminf _{n\to \infty }x_{n}:=\lim _{n\to \infty }{\Big (}\inf _{m\geq n}x_{m}{\Big )} \liminf _{n\to \infty }x_{n}:=\sup _{n\geq 0}\,\inf _{m\geq n}x_{m}=\sup\{\,\inf\{\,x_{m}:m\geq n\,\}:n\geq 0\,\}.$$

Why is it like this? Could anyone explain me in a very simple way this notion, even through a basic example? Thanks you very much for your help.

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    *I don't understand why should we consider a sequence of infima or suprema of the functions whose indices are greater or equal than a given number.* Do you think it should be something else instead? Usually, when somebody asks "why" something is defined a certain way, they have some other idea of how it should be defined – 2017-01-17
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    @Caro: If you have infinite sets $X := \{x_n : n \geq 0\}$ and $Y := \{x_n : n \geq 1\}$, consider $x := \inf(X)$ and $y := \inf(Y)$. What can you say about $x$ and $y$? You can say $x \leq y$. Do you understand why this is so? Since I assume you understand why, let us define $Z_m := \{x_n : n \geq m\}$. Then $\inf(Z_0) \leq \inf(Z_1) \leq \inf(Z_2) \leq ...$, that is, the $\inf(Z_m)$ are an **increasing** sequence of real numbers. What ist the limit of such a sequence? It is $\sup(\{\inf(Z_m): m \in \mathbb N\})$. – 2017-01-17
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    Thank you very much for your clear and quick reply. You talked about sets, does it works in the same way for functions too? I know my question is dumb and I'm sorry for this. – 2017-01-17
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    *Does it work. Could you explain me please how does this work for the sup? Thank you. – 2017-01-17
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    @Omnomnomnom: Yes, it is not clear at all the construction of these sequences of suprema and infima. Why the indices are considered to be greater or equal than m? – 2017-01-19

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The useful point of view, as I see it, is to consider $\limsup x_n $ as the greatest limit of any subsequence of $\{x_n\} $.

This is often useful in proofs that a limit exists; the advantage being that $\limsup $ and $\liminf $ exist for any bounded sequence.

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    Thank you for your reply. Could you provide me a more specific example? I'm newbie in all what refers to sequences of functions. I didn't get why we're considering a sequence of infima or suprema and say they increase/decrease respectively. I mean, I perfectly understand what does this mean with sets, but I can't imagine this for a sequence of functions... As an example, the sequence s(k)= x^k... Thank you – 2017-01-17