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**

Join Physics Forums Today!

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

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**