I was wondering whether the Bolzano-Weierstrass Theorem ("in a finite-dimensional normed space, every bounded sequence has a converging subsequence") would hold on
- a finite dimensional linear space
- with some obscure metric (rather than a norm)
As a counter example, I considered $\mathbb R$ with the metric $d(x,y)=\min\{1, |x-y|\}$ with the sequence $x_n:=n, n\in \mathbb N$ that is bounded (as a set in $\mathbb R$) but not convergent in any subsequence.
Is this correct so far? Are there more general considerations on this issue?
Best Answer
Your example works. You can also use the discrete metric in $\Bbb R^n$. Then every sequence is bounded, but only those with a constant subsequence have a convergent subsequence.
On the the other hand, asking whether it holds on a finite dimensional linear space makes no sense; such a space is not automatically endowed with a distance.