Cauchy Sequences Triangle Inequality.

  • Thread starter Enjoicube
  • Start date
1. The problem statement, all variables and given/known data
assuming an and bn are cauchy, use a triangle inequality argument to show that cn=
| an-bn| is cauchy

2. Relevant equations

an is cauchy iff for all e>0, there is some natural N, m,n>=N-->|an-am|<e

3. The attempt at a solution
I am currently trying to work backwards on this one, since we know
for some m,n>=N where N is a natural
Thus, through the triangle inequality this is >=|am-an+bn-bm|=|(bn-an)-(bm-am)|>=|bn-an|-|bm-am|.

Am I going completely wrong somewhere, because I am getting very stuck here. To anyone who answers, please only hints, not full answers (I am sure this is customary). Again, many thanks.

Tom Mattson

Staff Emeritus
Science Advisor
Gold Member
Two quick comments:

I would have said that [itex]|a_n-a_m|< \epsilon / 2[/itex] and [itex]|b_n-b_m|< \epsilon /2[/itex] just to get [itex]|(a_n-b_n)-(a_m-b_m)|<\epsilon[/itex].

Here's a nitpicky thing that your professor might correct (mine would have). Once [itex]\epsilon >0[/itex] is given, there's no reason to think that the same [itex]N[/itex] will work for both sequences. So I would have said that there exists [itex]N_1[/itex] for one sequence and [itex]N_2[/itex] for the other, such that blah blah blah... Then I would have let [itex]N=max\{N_1,N_2\}[/itex].
Yeah, my instructor does require this also. I can't believe the anguish this is causing me, It must be a very simple problem, and no one asked about it in class. Right now, I am supposed to be reading a history book, but this problem is back there bugging me.
Not to imply that anyone read this incorrectly, but the problem is cn=abs(an-bn) not just an-bn. Just entered my head that this might not have been emphasized.

Want to reply to this thread?

"Cauchy Sequences Triangle Inequality." You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Top Threads