Is d((xn), (yn)) = lim d(xn, yn) a metric for Cauchy sequences in (X, d)?

[Bachelier]

If (xn) and (yn) are two Cauchy sequences in a metric space (X, d), and we define
d((xn), (yn)) = lim d(xn, yn). Is this a metric on the set of all Cauchy sequences?

I'm thinking yes since all 3 properties work.

 

[Fredrik]

How do you check if the d you defined on the set of Cauchy sequences has the property $d(x,y)=0\Leftrightarrow x=y$?

 

[Bachelier]

thanks fredrik. I thought about it for a minute. So you're telling me it fails?

 

[Fredrik]

Yes, that's what I'm saying. You should try to find an example of two different Cauchy sequences S1 and S2 such that d(S1,S2)=0.

 

[Bachelier]

I'm thinking of a counter example something like:

let Sn = (0,1,0,1,0,1...) with nth term being 0, and Tn = (0,0,0,0...)

d(Sn,Tn)=lim d(sn, yn)= lim d(0,0)=0 but Sn is different from Tn.

what do you think?

 

[Fredrik]

0,1,0,1,0,1,... isn't a Cauchy sequence (and also isn't at 0 "distance" from 0,0,0,...).

 

[Bachelier]

0,1,0,1,0,1,... isn't a Cauchy sequence (and also isn't at 0 "distance" from 0,0,0,...).

nice,

let Sn= 1/n
Tn = 0,0,0,0,0

d(Sn, Tn) = lim d(sn, tn) = 0 yet sn does not equal tn.

 

[Fredrik]

Yes, that works. Another possibility is to let S be any Cauchy sequence, then change just one of its terms, and call the new sequence T. Then we have d(S,T)=0 but S≠T. For example, 1,1,1,1... and 0,1,1,1,...

 

[Bachelier]

Thank you Fredrik. :)