# Inner Limit in Normed Spaces by Open Balls

## Theorem

Let $\left \langle{C_n}\right \rangle_{n \in \N}$ be a sequence of sets in a normed vector space $\left({\mathcal X, \left\Vert{\cdot}\right\Vert}\right)$.

Then the inner limit of $\left \langle{C_n}\right \rangle_{n \in \N}$ is:

$\displaystyle \liminf_n \ C_n = \left\{{x: \forall \epsilon > 0: \exists N \in \mathcal N_\infty: \forall n \in N: x \in C_n + \epsilon B}\right\}$

where $B$ denotes the open unit ball of the space.

## 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.

$\blacksquare$