Question about the Bolzano Weierstrass analogue in Hilbert spaces

1. Dec 3, 2011

I<3Gauss

While reading a proof on the closure of the span of finite number vectors in a hilbert space with respect to the norm induced topology, I became stumped on a particular step of the proof using the Bolzano Weierstrass theorem.

For finite dimensional vector spaces, Bolzano Weierstrass states that:

"If ||an||<=M, then an contains a convergent subsequence (wrt whatever metric is defined on this vector space)"

Is it safe to assume that the above theorem extends to infinite dimensional vectors spaces (Hilbert spaces)?

Thanks!

2. Dec 3, 2011

micromass

No, consider the sequence

(1,0,0,0,...)
(0,1,0,0,...)
(0,0,1,0,...)
...

in $\ell^2$.

3. Dec 3, 2011

I<3Gauss

Thanks for the really simple example!