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I need to show that a metric space $X$ is complete iff for every nested sequence of closed balls in $X$ with radii tending to zero has nonempty intersection. A sequence $(B_n)$ of balls is said to be nested if $ B_{n}\supset B_{n+1}$ for all $n$.

Please help me get started. Thank you.

Daniel W. Farlow
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Klara
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2 Answers2

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$\newcommand{\cl}{\operatorname{cl}}$HINT: The easier direction is to show that if $X$ is complete, every nested decreasing sequence of closed balls has non-empty intersection. Let $\langle B_n:n\in\Bbb N\rangle$ be such a sequence, where $B_n$ is the closed ball of radius $r_n$ and centre $x_n$. You know that $\lim_{n\to\infty}r_n=0$; use this to prove that $\langle x_n:n\in\Bbb N\rangle$ is a Cauchy sequence, and that the limit of this sequence is in every $B_n$.

For the other direction, suppose that every nested sequence of closed balls has non-empty intersection, and let $\langle x_n:n\in\Bbb N\rangle$ be a Cauchy sequence. For $n\in\Bbb N$ let $T_n=\{x_k:k\ge n\}$, the $n$-th tail of the sequence. For $n\in\Bbb N$ let $d_n$ be twice the diameter of $T_n$, and let $B_n$ be the closed ball of radius $r_n=2d_n$ centred at $x_n$; then $T_n\subseteq B_n$.

Because the sequence is Cauchy, you know that $\lim_{n\to\infty}d_n=0$. Let $n\in\Bbb N$ be arbitrary. There is a $k>n$ such that $d_k<\frac12d_n$. Suppose that $y\in B_k$; then

$$d(y,x_n)\le d(y,x_k)+d(x_k,x_n)\le r_k+d_n=2d_k+d_n<2d_n=r_n\;,$$

so $y\in B_n$, and it follows that $B_k\subseteq B_n$. Use this observation to construct a nested subsequence of $\langle B_n:n\in\Bbb N\rangle$. By hypothesis this subsequence will have non-empty intersection $A$. Show that $A$ contains only one point, and show that $\langle x_n:n\in\Bbb N\rangle$ converges to this point.

Brian M. Scott
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Pick $x_n\in B_n$. What can you say about this sequence?

ncmathsadist
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  • Xn converges to zero, right? – Klara Feb 07 '13 at 03:34
  • @Klara: That doesn’t make sense: $0$ need not even be in $X$. In fact you can’t say much at all about the sequence unless the diameters of the sets $B_n$ tend to $0$; were you given that additional hypothesis? – Brian M. Scott Feb 07 '13 at 03:37
  • @ Brian The raddi of the balls tend to zero, doesnt that imply that the diameter of the sets B_n tend to 0? – Klara Feb 07 '13 at 03:39
  • @Klara: Yes, it does. Sorry: for some reason I didn’t see the hypothesis on the radii! – Brian M. Scott Feb 07 '13 at 03:41
  • @BrianM.Scott you are right, so I can say that Xn converges to x a point x s.t. d(x_n,x) goes to zero. – Klara Feb 07 '13 at 03:42
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    @Klara: You can show that $\langle x_n:n\in\Bbb N\rangle$ is Cauchy and then conclude that it has a limit. Then you have to show that the limit is in the intersection of the balls. When you’ve done that, you’ll have proved the $\implies$ direction of the theorem. – Brian M. Scott Feb 07 '13 at 03:44
  • @ Brian I will do that – Klara Feb 07 '13 at 03:45