Can someone help me come up with an example? (analysis)

[robertdeniro]

Homework Statement

let X be the set of all infinite dimensional vectors with finite nonzero components

consider 2 distance functions on X

d=euclidean distance=$$\sqrt{\sum(Xi-Yi)^{2}$$r=max distance=max{|X1-Y1|, |X2-Y2|, ...}

give an example of a sequence where it converges under r but not under d

Homework Equations

The Attempt at a Solution

i was thinking something like

v1={1, 0, ...}
v2={1, 1/2, 0, ...}
v3={1, 1/2, 1/4, 0, ...}

but it doesn't exactly work...

any ideas would be appreciated

 

[Tedjn]

Intuitively, when two vectors get close under r, each component gets close. When two vectors get close under d, an entire infinite sum must converge and decrease. Think of easy ways to make certain such an infinite sum does not converge -- how can we have this happen while still ensuring that components become arbitrarily close?

 

[Tedjn]

Ok. I realize I was probably too unclear. Here is an extra thought: suppose I have an infinite series that diverges, and I scale it by a positive factor less than 1. The series still diverges, but I have brought the terms closer to 0. Can you use this idea to build off what you were thinking about?

 

[robertdeniro]

are you trying to say

v1={1, 1, 1...}
v2=(1/2, 1/2, 1/2, ...}
and so on?

this won't work because remember there can only be finite number of non zero components

 

[Tedjn]

Not quite. I didn't phrase myself well in my first post; the second captures more the essence of what I am trying to say.

What I mean is that you can approximate the divergent series more and more (i.e. have one less zero in the infinite number of zeros at the end). You would also need to scale down each time so that the sequence converges to (0, 0, 0, ...) under r. Then, you would need to prove that such a sequence does not converge to anything under d. But that is the general idea.

Perhaps there is an easier example. I don't know.

 

[Dick]

Not quite. I didn't phrase myself well in my first post; the second captures more the essence of what I am trying to say.

What I mean is that you can approximate the divergent series more and more (i.e. have one less zero in the infinite number of zeros at the end). You would also need to scale down each time so that the sequence converges to (0, 0, 0, ...) under r. Then, you would need to prove that such a sequence does not converge to anything under d. But that is the general idea.

Perhaps there is an easier example. I don't know.

{1,1/2,1/4,1/8,...} is a bad example because it's in both vector spaces since it's bounded AND square summable. Find a sequence that is bounded but not square summable, is what Tedjn is trying to say. And then repeat your first attempt using that instead.