I posted this question many years ago on math stackexchange but it did not get an answer. It had circulated as a puzzle in graduate school.


A disk $D$ of radius $1$ contains disks $D\_i$ ($i \ge 1$) of radius $r\_i<1$ with pairwise disjoint interiors. Assuming the $D\_i$ "use up" the area of $D$ in the sense that $\sum r\_i^2=1,$ show the sum of the *unsquared* radii $\sum r\_i$ diverges.



On covering a disk by non-overlapping subdisks
Proven by [O. Wesler, “An infinite packing theorem for spheres,” PAMS Vol. 11, pp. 324-326, (1960).](https://sci-hub.do/10.2307/2032977)


