I've looked at the Wikipedia article, but it seems like gibberish. The only thing I was able to pick out of it was the concept of infimum (greatest lower bound) and supremum (least upper bound), as I had learned them previously in an intro discrete math course.
The limit inferior of a sequence ($x_n$) is defined by
$\displaystyle\liminf_{n\to\infty}x_n := \lim_{n\to\infty}\Big(\inf_{m\geq n}x_m\Big)$
or
$\displaystyle\liminf_{n\to\infty}x_n := \sup_{n\geq 0}\,\inf_{m\geq n}x_m=\sup\{\,\inf\{\,x_m:m\geq n\,\}:n\geq 0\,\}.$
Similarly, the limit superior of ($x_n$) is defined by
$\displaystyle\limsup_{n\to\infty}x_n := \lim_{n\to\infty}\Big(\sup_{m\geq n}x_m\Big)$
or
$\displaystyle\limsup_{n\to\infty}x_n := \inf_{n\geq 0}\,\sup_{m\geq n}x_m=\inf\{\,\sup\{\,x_m:m\geq n\,\}:n\geq 0\,\}.$
Can anybody provide any examples of its use, and why it's used in that context?
Best Answer
I've found that some students have difficulty understanding the usual definitions of limit superior and inferior because these definitions combine the notions of limits, of suprema, and of infima, all of which the student may have learned only recently and not fully internalized. For such students, I like to give the following alternative definitions, equivalent to the usual ones but not containing the words "limit", "supremum", and "infimum". (Nor are there absolute values or visible $\varepsilon$'s.)
A number $t$ is the limit superior of a sequence $\langle a_n\rangle$ if the following two conditions are both satisfied:
For every $s<t$ we have $s<a_n$ for infinitely many $n$'s.
For every $s>t$ we have $s<a_n$ for only finitely many $n$'s (possibly none).
Similarly, a number $t$ is the limit inferior of a sequence $\langle a_n\rangle$ if the following two conditions are both satisfied:
For every $s>t$ we have $s>a_n$ for infinitely many $n$'s.
For every $s<t$ we have $s>a_n$ for only finitely many $n$'s (possibly none).
Two additional remarks may be useful:
The definition of lim inf is gotten from the definition of lim sup by simply reversing all inequalities.
The definitions can be easily extended to $\pm\infty$ in place of numbers $t$. Just adopt the convention that, even then, $s$ refers to actual numbers, all of which are $>-\infty$ and $<+\infty$.