This seems to be a fairly easy question but I'm looking for new points of view on it and was wondering if anyone might be able to help
(by the way- this question does come from home-work, but I've already solved and handed it, and I'm posting this out of interest, so no HW tag)
Let $B_n=B(x_n,r_n)$ be a sequence of nested closed balls in a Banach space $X$ prove, that $\bigcap_1^\infty B_n\neq\emptyset$
As I said before, it should be rather simple. When the radii decrease to 0, it's just a matter of selecting any sequence of points in $B_n$, and it must be Cauchy- and the limit is in the intersection.
My question is what to do when the radii do not decrease to 0? I got some tips about multiplying the balls by a sequence of decreasing scalars, or reducing the radii so that they decrease to 0, but found too many pathological cases for both methods.
Finally- I used a geometric arguemnt (which i've shown to work in any normed space) that if $B(x_1,r_1)\subset B(x_2,r_2)$ then $\| x_1-x_2\|\leq|r_1-r_2|$. This turned out to be some kind of technical catastrophe, but it worked...
Still, if anyone knows of a more elegant solution, I'd love to hear about it