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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 $\bigcap_{n=1}^\infty B_n\neq\varnothing$.

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.

Tomasz Kania
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kneidell
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  • What's so catastrophic about the geometric argument? All you need is that the affine line spanned by $x_1$ and $x_2$ is isometrically isomorphic to the ground field, then it's just a 1d (or 2d if your space is complex) picture. – Chris Eagle Feb 17 '11 at 09:27
  • "[...] be a sequence of nested in a banach space". Seems like you missed something there? Should it be "nested closed balls"? – kahen Feb 17 '11 at 09:50
  • yeah, it's nested sequence- as the title suggests :) i'm changing it – kneidell Feb 17 '11 at 09:56
  • It's not true in general. That property is called spherical completeness and fails for the p-adic complex numbers, for example. – George Lowther Feb 17 '11 at 10:05
  • ... But that is a complete metric space, rather than a Banach space – George Lowther Feb 17 '11 at 10:15
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    @George: Perhaps the simplest example of a complete metric space where this fails is the natural numbers with the metrc $d(m,n)=1+1/(\min (m,n))$. – Chris Eagle Feb 17 '11 at 11:35

1 Answers1

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I don't know if this is more elegant, but that's about the best I can come up with at the moment and probably essentially the same as your argument.

Consider first the situation $B_{\leq r}(x) \subset B_{\leq s}(y)$. It is easy to see that $r \leq s$.

Claim. $\|y - x\| \leq s - r$.

Proof. If $x = y$ there is nothing to prove, so let's assume $x \neq y$. The point $z = x - r \frac{y-x}{\|y - x\|}$ belongs to $B_{\leq r}(x)$ and hence also to $B_{\leq s}(y)$. Therefore $\|y - z\| \leq s$. On the other hand, \[ y - z = y - x + \frac{r}{\|y - x\|} (y - x) = \underbrace{\left(1 + \frac{r}{\|y - x\|}\right)}_{\lambda} (y - x), \] so $s \geq \lambda \|y - x\| = \|y - x\| + r$ and hence $\|y - x\| \leq s - r$.

This means that a nested sequence of closed balls $B_{\leq r_{n}}(x_{n})$ has the following properties:

  1. The sequence $r_{n}$ is monotonically decreasing, hence converges to some $r$.
  2. If $N$ is such that $r_{N} \leq r + \varepsilon$ then the above claim implies that for all $n\geq m \geq N$ we have $r_m - r_n \leq \varepsilon$, so $\|x_{m} - x_{n}\| \leq \varepsilon$ because $B_{\leq r_{n}}(x_{n}) \subset B_{\leq r_{m}}(x_{m})$.

In other words, the centers $x_{n}$ form a Cauchy sequence and their limit point $x$ must belong to $\bigcap_{n = 1}^{\infty} B_{\leq r_{n}}(x_{n})$.

Added: As Jonas pointed out, the argument can be made even simpler and doesn't need completeness: Suppose $r_{n} \to r \gt 0$. Then there is $N$ such that $r_{N} \leq 2r$. Then for all $n \geq N$ we have $r \leq r_{n} \leq r_{N} \leq 2r$, so $r_{N} - r_{n} \leq r$ and the claim implies that $\|x_{n} - x_{N}\| \leq r \leq r_{n}$, so $x_{N} \in \bigcap_{n = 1}^{\infty} B_{\leq r_{n}} (x_{n})$.

t.b.
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  • @kneidell: While writing this I have forgotten about the fact that you've already had this solution. I don't think there is a much easier way. – t.b. Feb 17 '11 at 12:19
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    If the limit $r$ of the radii is positive, then $r_N<2r$ for some $N$, and then $x_N$ will be in every subsequent ball (by the claim). – Jonas Meyer Feb 20 '11 at 04:33
  • @Jonas: Right. So you needn't even use completeness in that case. Nice! This hasn't occurred to me before, thanks! – t.b. Feb 20 '11 at 04:47
  • Did you nowhere use the reflexivity of the Banach space? What about this example? – superAnnoyingUser Jul 04 '13 at 09:16
  • Sorry for this comment which is probably stupid but I am wondering how you would prove that $B_r(x) \subset B_s(x)$ implies $r \le s$. What about these examples: http://math.stackexchange.com/questions/734248/example-of-two-open-balls-such-that-the-one-with-the-smaller-radius-contains-the ? Thanks. – Romeo Aug 09 '14 at 10:31
  • @Romeo:Those examples are not normed spaces. Suppose $B_r(x)\subseteq B_s(y)$. If $x=y$, then we are saying that $|x-z|\leq r$ implies $|x-z|\leq s$, and this implies $r\leq s$ (all that is needed for this part are that there are vectors at distance exactly $r$ from $x$, easy to show for any nonzero normed space). If $x\neq y$, then $x+r\frac{x-y}{|x-y|}$ is in $B_r(x)$, hence also in $B_s(y)$. This implies that $|x-y|+r\leq s$, so $r<s$. – Jonas Meyer Aug 14 '14 at 08:09
  • @Student: (I know you asked over a year ago, don't know if this is helpful.) Nowhere in the question is the Banach space assumed to be reflexive, nor is it needed. But the balls in a normed space are special compared to balls in a general metric space. The answer uses the fact that the metric comes from the norm. – Jonas Meyer Aug 14 '14 at 08:12
  • @Romeo It's trivial that if $E\subset F$ in any metric space then the diameter of $E$ is no larger than the diameter of $F$. Now if we're talking about balls in a normed vector space the radius is half the diameter. – David C. Ullrich Apr 22 '16 at 14:26