About square summable sequences space

[antiņš]

First, I'm sorry for my bad english.

Homework Statement

I need to disprove:
$$(x_n) \in \ell^2$$ is a Cauchy sequence, if $$\displaystyle \lim_{x \to \infty} d(x_n, x_{n+1})=0$$.

Homework Equations

Ok, sequence is Cauchy sequence if $$\exists n_0 \; \forall p,q>0 \; d(x_p,x_q) \rightarrow 0$$

The Attempt at a Solution

Has someone idea about this? I tried 1/ln(x) and many examples like this one, but all this are wrong.

 

[HallsofIvy]

First, I'm sorry for my bad english.

Homework Statement

I need to disprove:
$$(x_n) \in \ell^2$$ is a Cauchy sequence, if $$\displaystyle \lim_{x \to \infty} d(x_n, x_{n+1})=0$$.

Homework Equations

Ok, sequence is Cauchy sequence if $$\exists n_0 \; \forall p,q>0 \; d(x_p,x_q) \rightarrow 0$$

You mean $$\forall p,q> n_0$$

The Attempt at a Solution

Has someone idea about this? I tried 1/ln(x) and many examples like this one, but all this are wrong.

Here's a hint: $$\sum_{n=1}^\infty\frac{1}{n}$$ does not converge.

Your English is excellent. (Well, except for not capitalizing "English"!)

 

[antiņš]

Of course, $$x_n=(1, \frac{1}{2}, ... , \frac{1}{\sqrt{n}})$$ is what I'm looking for. And this sequence is square summable because $$x_n$$ is finite.

Thanks! ^_^

 

[kbaumen]

First, I'm sorry for my bad english.

No worries mate, Latvians speak English well.