You are Here: Home >< Maths

# Need help with defining a metric watch

1. Kind of a trivial question, but if you have the sequence space , endowed with the metric

How do you define ?
2. (Original post by RamocitoMorales)
Kind of a trivial question, but if you have the sequence space , endowed with the metric

How do you define ?
It's not clear what you're asking here: how are you taking the sequence to differ from the sequence ?
3. (Original post by Gregorius)
It's not clear what you're asking here: how are you taking the sequence to differ from the sequence ?
I define for a Cauchy sequence.
4. (Original post by RamocitoMorales)
I define for a Cauchy sequence.
So here you have appearing to choose fixed constant integers m and n greater than some particular integer, but in the next you're using n as the dummy variable indexing your sequence.Confusing!

Could you give an example to illustrate what you are asking?
5. (Original post by Gregorius)
Could you give an example to illustrate what you are asking?
The greater picture is that I'm trying to prove that the sequence space , endowed with the metric

is complete.

A metric space is complete if all Cauchy sequences are convergent.

I let be a Cauchy sequence, then such that ,

I was having trouble defining . Is that more clear?

Spoiler:
Show
Fix natural numbers if that makes it less confusing.
6. (Original post by RamocitoMorales)
The greater picture is that I'm trying to prove that the sequence space , endowed with the metric

is complete.
Right, I can see what you are trying to do now. The problem is that each "point" in the sequence space is a sequence of elements of your underlying field: . You need to consider a cauchy sequence of these "points", where each "point" is itself a sequence.. In other words, a sequence of sequences.
7. (Original post by Gregorius)
Right, I can see what you are trying to do now. The problem is that each "point" in the sequence space is a sequence of elements of your underlying field: . You need to consider a cauchy sequence of these "points", where each "point" is itself a sequence.. In other words, a sequence of sequences.
But then if we take which is another sequence and not convergent to a point;
8. (Original post by RamocitoMorales)
But then if we take which is another sequence and not convergent to a point;
Since X is a sequence space, a point in X is a sequence.

More generally, in problems like this, it is desperately important that you are clear about notation, because you'll need to distinguish between sequences as opposed to sequences of sequences. That's why Gregorius was explictly using bold face for elements of X - they're elements, but they are still sequences (over .

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: September 11, 2015
Today on TSR

### He lied about his age

Thought he was 19... really he's 14

### University open days

Wed, 25 Jul '18
2. University of Buckingham
Wed, 25 Jul '18
3. Bournemouth University
Wed, 1 Aug '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams