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I have a little doubt. These quantities

$\inf_{n} f_n $ , $\sup_n f_n$

are function only if the there exist the infimum of {$f_n(x)$} $\forall x $ and is different from $\infty$.

And the same is for the superior and inferior limit.

Am I right?

For instance, for the sequence of constant function $f_n(x) = n$ we can't define $\sup_n f_n $ right?

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    I'm a little confused by what you mean by $\inf_n$, what is $n$ supposed to mean? – 2017-02-18
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    In your example, if $n = 1, 2, 3, \dots$, then we would have $\inf_n f_n(x) = 1$. – 2017-02-18
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    I meant $sup_n f_n$ – 2017-02-18
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    We can , it's just the function $f(x) = +\infty$. In measure theory we usually work in the extended reals $[-\infty, +\infty]$. – 2017-02-18
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    Henno Brandsma, thanks! So we treat $\infty$ as a real number? Do you have any pdf explaining this? – 2017-02-18

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