A sequence [itex]\{x_n\}[/itex] in a metric space [itex](X,d)[/itex] converges iff(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

(\exists x\in X)(\forall \epsilon > 0)(\exists N \in \mathbb N)(\forall n > N)(d(x_n,x) < \epsilon).

[/tex]

Am I correct when I assert that the negation of this is: A sequence [itex]\{x_n\}[/itex] doesnotconverge in [itex](X,d)[/itex] iff

[tex]

(\forall x\in X)(\exists \epsilon > 0)(\forall N \in \mathbb N)(\exists n > N)(d(x_n,x) \geq \epsilon)?

[/tex]

So, if I'm trying to show a sequence does not converge, I let [itex]x\in X[/itex] be given and show that there is some [itex]\epsilon[/itex] neighborhood of this point that contains at most finitely many of the [itex]x_n[/itex]?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Showing a sequence is divergent

**Physics Forums | Science Articles, Homework Help, Discussion**