Proof of "Every Cauchy Sequence is Bounded

[fderingoz]

I read the proof of the proposition "every cauchy sequence in a metric spaces is bounded" from

http://www.proofwiki.org/wiki/Every_Cauchy_Sequence_is_Bounded

I don't understand that how we can take m=N$_{1}$ while m>N$_{1}$ ?

In fact i mean that in a metric space (A,d) can we say that

[$\forall$m,n>N$_{1}$$\Rightarrow$ d(x$_{n}$,x$_{m}$)<1]$\Rightarrow$[$\forall$n$\geq$N$_{1}$$\Rightarrow$ d(x$_{n}$,x_{$_{N_{1}}$})<1]

 

[Erland]

I read the proof of the proposition "every cauchy sequence in a metric spaces is bounded" from

http://www.proofwiki.org/wiki/Every_Cauchy_Sequence_is_Bounded

I don't understand that how we can take m=N$_{1}$ while m>N$_{1}$ ?

In fact i mean that in a metric space (A,d) can we say that

[$\forall$m,n>N$_{1}$$\Rightarrow$ d(x$_{n}$,x$_{m}$)<1]$\Rightarrow$[$\forall$n$\geq$N$_{1}$$\Rightarrow$ d(x$_{n}$,x_{$_{N_{1}}$})<1]

You are right. This is an error in the wiki. $m,n>N$ should be changed to $m,n\ge N$ wherever it occurs (this also holds with $N_1$ instead of $N$). This fits the wiki's definition of Cauchy sequence, which the wiki's proof doesn't.

 

[fderingoz]

Thank you for the answer i also think like you. This is an error in the wiki. But i saw several functional analysis book which write the proof of proposition same as in wiki. So,

Who is wrong?

 

[Erland]

Well, we can define "Cauchy sequence" with either $>$ or $\ge$, but in the former case, we cannot use $N_1$ the way it is used in the proof in the wiki. Then we also need an $N_2>N_1$ to work with, or something like that.