Inner Limit in Normed Spaces by Open Balls
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Theorem
Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in a normed vector space $\struct {\XX, \norm {\, \cdot \,} }$.
Then the inner limit of $\sequence {C_n}_{n \mathop \in \N}$ is:
- $\ds \liminf_n C_n = \set {x: \forall \epsilon > 0: \exists N \in \NN_\infty: \forall n \in N: x \in C_n + B_\epsilon}$
where $B_\epsilon$ denotes the open $\epsilon$-ball of the space.
 | This article, or a section of it, needs explaining. In particular: What are $N$ and $\NN_\infty$ in this context? Also, what point is at the center of $B$? And what does $C_n + \epsilon B$ mean? For the latter one suspects $\cup$, but this needs to be checked. This page might need to be rewritten from a new perspective, as the original author was touchy about symbols used and departed $\mathsf{Pr} \infty \mathsf{fWiki}$ in a rage when his notation was changed. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Proof
The proof is an immediate result of Inner Limit in Hausdorff Space by Open Neighborhoods since the arbitrary open sets can be here replaced by open balls.
$\blacksquare$