TITLE: Does Cauchy sequence of disjoint and closed subsets converge to a non-empty set in a Banach space?
QUESTION [6 upvotes]: Suppose, $X$ is a banach space. For any $x,y \in X$, we define $d(x,y) = |x-y|$, For any $A, B \subseteq X$, we define:
$$d(A, B) = \inf_{x \in A, y\in B}{d(x,y)}$$
Say,$(K_n)_{n \in \mathbb{N}}$ is a Cauchy sequence of disjoint and closed subsets of$X$. We say that:
$$K = \{x \in X: \lim_{n \to \infty}d(\{x\}, K_n) = 0\}$$ is the limit of $(K_n)_{n \in \mathbb{N}}$.
Is $K$ necessarily non-empty?
Added: Thank you all for your examples and corrections. My naive attempt is to find a condition stronger than sequential completeness. I hope it will work by adding "disjoint and closed".

REPLY [3 votes]: Taking my comment, adjusting to make it closed and disjoint.  In the plane, let $A_0, A_1, A_2$ be the three sides of an equilateral triangle, all of length $1$.  They are closed but not disjoint.  Define sequence $K_n$ as follows:  If $n=3m$, let $K_n$ be $A_0$ moved radially outward by distance $1/n$.  If $n=3m+1$, let $K_n$ be $A_1$ moved radially outward by distance $1/n$.  If $n=3m+2$, let $K_n$ be $A_2$ moved radially outward by distance $1/n$.  Then this is a Cauchy sequence, $d(K_i,K_j) \le 1/i+1/j$,
but its "limit" is empty.