Let $A$ be a bounded region in $\mathbb{R^2}$ and $C_n={(K_n,r_n)}$ a sequence of circles of decreasing radius that do not intersect, such that $C_{i+1}$ is externally tangent to $C_i$ $\forall i\in\mathbb{N}$ and $K_1=(x_1, y_1)$ $\in A$ with $C_1$ not covering $A$.
Can such a sequence of circles cover any bounded region of $\mathbb{R^2}$, and if not what are the properties of the bounded regions that it can cover?
Excluding $C_1$, we have full freedom in choosing the centers and radii of the circles that form the sequence as long as they respect the conditions mentioned.
Excluding trivial cases where, for example the region is itself a circle (where we can simply choose this as $C_1$ and end the sequence there), or the shape of the region corresponds identically to that which would be produced by a finite number of $C_i$, my thoughts on the problem-besides coming up with it-I fear are rather limited.
I tried to think of it in terms of lines and line segments but the analogy is very weak if non-existent. "Obvious" generalizations can be conceived (a sequence of spheres on $\mathbb{R^3}$ covering bounded volumes) but as we go up in terms of dimensions it becomes far more complicated and any intuition is lost.
I think it has to do with how "smooth" the boundary of the region is in order for the sequence to approach it but in strict mathematical terms I have to admit I am not familiar with a relative concept that I could employ here.