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Set limes inferior
Definition
| $A\in \text{Seq}(X)$ |
| $\underset{n\to\infty}{\liminf}A_n$ |
| ${\bigcap_{n=1}^\infty}\left({\bigcup_{k=n}^\infty}A_n\right)$ |
Ramifications
We have that
$\underset{n\to\infty}{\limsup}A_n\subseteq \underset{n\to\infty}{\liminf}A_n,$
see set limes superior. If moreover
$\underset{n\to\infty}{\limsup}A_n=\underset{n\to\infty}{\liminf}A_n,$
then we call it
$\underset{n\to\infty}{\lim}A_n$
and say $A$ is convergent.
Reference
Wikipedia: Limit superior and limit inferior
Discussion
For a more possibly more elucidating explaination, see my answer in this Math.Se thread this Math.SE thread
Context
Parameterized set
Parameter refinement of
Parents
