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I am wondering about the following question:

Given a (countable) sequence of nested open balls:

$$ B_1 \supseteq B_2 \supseteq \cdots $$ Not necessarily having the same same center. All having radius bounded from below, say by $r > 0$. Then can we say that $$\bigcap_{i=1}^{\infty} B_i \neq \varnothing$$ it is certainly true in the reals, as one can simply go to the point where the radii are close to $r$ then take the center of that ball. However, I'm having trouble seeing whether or not it is true in the general case. So thanks in advance for proof or counterexample.

Deven Ware
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2 Answers2

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Let $N$ be the set of positive integers. Define a metric on $N$ as follows: $$d(m,n)=\left\{\begin{array}{ll} 1+ \frac{1}{mn}& \mbox{if }m \neq n\\ 0 &\mbox{if }m=n \end{array}\right.$$ It is straightforward to check that this is a complete metric on $N$ (complete because this metric is discrete).

The closed balls $$B(n, 1+1/n^2)=\{m: m\ge n\}, n=1,2,\cdots$$ are decreasing and have empty intersection. Of course, the corresponding open balls also have empty intersection.

TCL
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The below argument is incorrect as pointed out by TCL. I'll leave the answer here anyway, just incase.

Note that $\bigcap_i^n B_i=B_n$ because each $B_i$ is a subset of $B_{i+1}$. Let $d_i=\mbox{diam}(B_i)$. It should be clear that if $\bigcap_i^{\infty} B_i=\emptyset$, then $\lim_{i\rightarrow\infty}d_i=0$ and so there exists some $i\in\mathbb{N}$ such that $d_i< 2r$, but then $B_i$ is an open ball of radius less than $r$ which contradicts the assumption that each $B_i$ has radius bounded below by some non-zero $r$.

I should add that i'm using the definition of the diameter of a subset $A$ of a metric space $X$ with metric $d$ to be $\mbox{diam}(A)=\sup\{d(x,y)|x,y\in A\}$.

Dan Rust
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    See my answer above. – TCL May 12 '13 at 00:14
  • Hmm, any idea where my argument falls down? – Dan Rust May 12 '13 at 00:20
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    "It should be clear that if $\bigcap_i^{\infty} B_i=\emptyset$, then $\lim_{i\rightarrow\infty}d_i=0$" --this is the problem. – TCL May 12 '13 at 00:26
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    It seems to be a general principle that, when looking for the error in an argument, the first place to look is wherever words like "clear", "obvious", "routine", or "trivial" occur. – Andreas Blass May 12 '13 at 00:28
  • Yes, I guess one should always be careful when assuming that the property of a limit is the limit of the property. – Dan Rust May 12 '13 at 00:31