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- Thread starter Amok
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What is a sequence that should converge? Well a sequence who's terms lie closer and closer together. For example, the sequence (1/n) should converge, because the terms are closer and closer. But (n) does not converge, because the terms both have distance 1 from each other.

The space [tex]\mathbb{R}[/tex] is complete: every sequence that should converge converges, but [tex]\mathbb{Q}[/tex] is incomplete, indeed a rational sequence that converges to [tex]\pi[/tex] does not converge in [tex]\mathbb{Q}[/tex].

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What do you mean with "a Hilbert spaces converges"?But cam you show that a Hilbert space or a [tex]\mathbb{R}[/tex] space converges? Using the definition of distance, for example?

Yes, one can show that for a lot of spaces, so it's certainly not an impossible condition to check. The only space for which it is really hard to check is for [tex]\mathbb{R}[/tex], but that's because the definition of [tex]\mathbb{R}[/tex] is quite complicated...Can you show that every Cauchy sequence in a certain space converges?

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