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.

$\blacksquare$