Question about the Bolzano Weierstrass analogue in Hilbert spaces

[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!

 

[micromass]

No, consider the sequence

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

in \ell^2.

 

[I<3Gauss]

Thanks for the really simple example!