Inner Limit of Sequence of Sets in Normed Space

Theorem
Let $\struct {\XX, \norm {\, \cdot \, } }$ be a normed vector space.

Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $\XX$.

The inner limit of $\sequence {C_n}_{n \mathop \in \N}$ is:
 * $\ds \liminf_n C_n = \set {x \in X: \lim_n \map d {x, C_n} = 0}$

where $d$ stands for the point-to-set distance mapping.

Proof
$(1): \quad$ We need to show that:
 * $\ds \limsup_n \ \map d {x, C_n} = 0$

that:
 * $\ds \limsup_n \ \map d {x, C_n} > 0$

that is, that there exists an increasing sequence of indices $\sequence {n_k}_{k \mathop \in \N}$ so that $\map d {x, C_{n_k} } \to_k a > 0$.

This suggests that there exists a $\epsilon_0 \in \R_{>0}$ such that:
 * $\forall k \in \N: \map d {x, C_{n_k} } > \epsilon_0$

However, according to Inner Limit in Hausdorff Space by Set Closures:


 * $\ds x \in \map \cl {\bigcup_{k \mathop \in \N} C_{n_k} }$

while:
 * $\ds \map d {x, \map \cl {\bigcup_{k \mathop \in \N} C_{n_k} } } \ge \epsilon_0$

which is a contradiction.

Hence:
 * $\ds \limsup_n \map d {x, C_n} = 0$

That is:
 * $\ds \lim_n \map d {x, C_n} = 0$

and thus we have proven that $x$ is in the set $\ds \set {x \in X: \lim_n \ \map d {x, C_n} = 0}$.

$(2): \quad$ Let $\ds x \in \set {x \in X : \lim_n \ \map d {x, C_n} = 0}$.

This is:
 * $\ds \lim_n \map d {x, C_n} = 0$

For any $\epsilon > 0$, we can find $n_0 \in \N$ such that:
 * $\forall n \ge n_0: \map d {x, C_n} \le \dfrac \epsilon 2$

By definition:
 * $\map d {x, C_n} = \inf \set {\norm {x - y}: y \in C_n}$

Thus we can find a $y_n \in C_n$ such that:


 * $\norm {y_n - x} < \map d {x, C_n} + \dfrac \epsilon 2 = \epsilon$

That is:


 * $\exists y_n \in C_n: \norm {y_n - x} < \epsilon$

Therefore:
 * $x \in C_n + \epsilon \BB$

where $\BB$ is the open unit ball of $\XX$.

That is:
 * $\BB := \set {x: \norm x < 1}$

Thus:
 * $\epsilon \BB = \set {x: \norm x < \epsilon}$

where $C_n + \epsilon \BB$ denotes the Minkowski sum of $C_n$ and $\epsilon \BB$.

From this observation, it follows from Inner Limit in Normed Spaces by Open Balls that:
 * $\ds x \in \liminf_n C_n$

Also see

 * Inner Limit in Hausdorff Space by Open Neighborhoods