Are Convergence Sequences Unique in a Normed Linear Space?

[bugatti79]

Folks,

Can anyone give an example of where convergence sequences can be unique in a normed linear space?

THanks

 

[micromass]

What do you mean with "convergence sequences are unique"?? I don't really understand what you're asking.

 

[bugatti79]

What do you mean with "convergence sequences are unique"?? I don't really understand what you're asking.

if x_n is a sequence and converges to x and y then x=y. THis is what I am looking for.

 

[micromass]

if x_n is a sequence and converges to x and y then x=y. THis is what I am looking for.

Yes, that's true in every normed space. What are you looking for?? A proof of that?

 

[bugatti79]

Yes, that's true in every normed space. What are you looking for?? A proof of that?

yes if it is standard textbook stuff. Its not in my book. Thanks !

 

[micromass]

OK, I'll give you a hint on how to prove it:

$$\|x-y\|\leq \|x-x_n\|+\|x_n-y\|$$

Use the definition of convergence of a sequence now.

 

[bugatti79]

OK, I'll give you a hint on how to prove it:

$$\|x-y\|\leq \|x-x_n\|+\|x_n-y\|$$

Use the definition of convergence of a sequence now.

How this you arrive with the right hand side of the expression? Would the line before that have been

$$\|x-y\|\leq \|(x-x_n)+(x_n-y)\|$$

 

[dextercioby]

Yes, but with the =[/size] sign, since you're adding and subtracting the same number inside the norm.

 

[bugatti79]

OK, I'll give you a hint on how to prove it:

$$\|x-y\|\leq \|x-x_n\|+\|x_n-y\|$$

Use the definition of convergence of a sequence now.

ok,

$$\|x-y\|\leq \|(x-x_n)+(x_n-y)\|$$

$$\|x-y\|= \|(x-x_n)+(x_n-y)\| \le \|x-x_n\|+\|x_n-y\| \le \|-(x_n-x)\|+\|x_n-y\|$$

I have rewritten the above because we need the expressions in the form ||xn-x||=0 as n tends to infinity

now,
since xn=x for the limit n to infinity and and given ε>0 then ε/2>0

therefore so there exist $$n_1 ∈ N$$ such that

$$\|-(x_n-x)\|<ε/2$$ and $$n_2 ∈ N$$

$$\|(x_n-y)\|<ε/2$$

How I doing so far?

 

[bugatti79]

ok,

$$\|x-y\|\leq \|(x-x_n)+(x_n-y)\|$$

$$\|x-y\|= \|(x-x_n)+(x_n-y)\| \le \|x-x_n\|+\|x_n-y\| \le \|-(x_n-x)\|+\|x_n-y\|$$

I have rewritten the above because we need the expressions in the form ||xn-x||=0 as n tends to infinity

now,
since xn=x for the limit n to infinity and and given ε>0 then ε/2>0

therefore so there exist $$n_1 ∈ N$$ such that

$$\|-(x_n-x)\|<ε/2$$ and $$n_2 ∈ N$$

$$\|(x_n-y)\|<ε/2$$

How I doing so far?

$$n_1 ∈ N$$

The third line back up...I think I can take out the minus sign giving


$$-\|(x_n-x)\|<ε/2$$ and $$n_2 ∈ N$$

$$\|(x_n-y)\|<ε/2$$...Any suggestions on what to do next?

 

[Fredrik]

since xn=x for the limit n to infinity and and given ε>0 then ε/2>0

therefore so there exist $$n_1 ∈ N$$ such that

$$\|-(x_n-x)\|<ε/2$$ and $$n_2 ∈ N$$

$$\|(x_n-y)\|<ε/2$$

How I doing so far?

Not bad, but there are some inaccuracies. You should start by saying something like "let ε>0 be arbitrary". Then "Since $\lim x_n=x$, there's an $n_1\in\mathbb N$ such that..." You also left out something really essential in that step. You didn't say that the first inequality holds for all $n\in\mathbb N$ such that $n\geq n_1$.

The third line back up...I think I can take out the minus sign giving


$$-\|(x_n-x)\|<ε/2$$ and $$n_2 ∈ N$$

Look at the definition of "norm" again.

$$\|(x_n-y)\|<ε/2$$...Any suggestions on what to do next?

The same thing we discussed in the other thread. This step is part of every proof that involves the definition of limits of sequences and the triangle inequality.

 

[bugatti79]

Ok, the norm means the evaluation will be strictly positive hence it is ok to leave the minus sign inside...I think! here goes..

Let $\varepsilon>0$ be arbitrary. Let $n_1\in\mathbb N$ be such that for all $n\in\mathbb N$, $$n\geq n_1\ \Rightarrow\ \|-(x_n-x)\|<\frac{\varepsilon}{2}.$$ Let $n_2\in\mathbb N$ be such that for all $n\in\mathbb N$, $$n\geq n_2\ \Rightarrow\ \|x_n-y\|<\frac{\varepsilon}{2}.$$ For all $n\in\mathbb N$ such that $n\geq \max\{n_1,n_2\}$,

$$||(x-y)\| \leq ||-(x_n-x)||+||x_n-y|| < \frac{\varepsilon}{2}+\frac{\varepsilon}{2}< \varepsilon$$...?

 

[Fredrik]

Yes, that's correct. As you have already noticed, the proof is essentially the same as in the other thread. This is a standard proof method that you will undoubtedly have to use many times again in the future.

What i wanted you to see when I said that you should look at the definition of "norm" again, is that the definition includes the condition $\|ax\|=|a|\,\|x\|$. You should use this to get rid of that minus sign.

Now there are two more things you have to make sure that you understand before this proof is complete. The first is that if t is a non-negative real number, and t<ε for all ε>0, then t=0. The second is that the only vector that has norm 0 is the zero vector.

 

[bugatti79]

Yes, that's correct. As you have already noticed, the proof is essentially the same as in the other thread. This is a standard proof method that you will undoubtedly have to use many times again in the future.

What i wanted you to see when I said that you should look at the definition of "norm" again, is that the definition includes the condition $\|ax\|=|a|\,\|x\|$. You should use this to get rid of that minus sign.

Now there are two more things you have to make sure that you understand before this proof is complete. The first is that if t is a non-negative real number, and t<ε for all ε>0, then t=0. The second is that the only vector that has norm 0 is the zero vector.

Ok, thanks! Where would t come into play in this problem or did you mean n?

Thanks in advance!

 

[Sina]

more generally it is valid in haussdorf spaces because you can separate each point with two nbds which both has to contain all but finite number of the sequence hence contradictions. It will might also be enough for it to be one of those T something spaces whose definitions I don't remeber now.

 

[Fredrik]

Ok, thanks! Where would t come into play in this problem or did you mean n?

I'm saying that you need the following theorem:

For all t≥0, if t<ε for all ε>0, then t=0.​

It's a "for all" statement, so it doesn't matter what symbol we use. Note that it's quite common to express "for all" statements as "if" statements. For example, "for all real t, t²≥0", can be expressed as "if t is real, then t²≥0". These statements should actually both be considered abbreviated forms of the more accurate "for all t, if t is a real number, then t²>0".

 

[dextercioby]

more generally it is valid in haussdorf spaces because you can separate each point with two nbds which both has to contain all but finite number of the sequence hence contradictions. It will might also be enough for it to be one of those T something spaces whose definitions I don't remeber now.

T(0,1,2,3,4) comes from separation properties. Hausdorff is T2. I suspect* the proof you gave also requires first-countability.

* somebody check!

 

[Sina]

Why does it require first countability?

I can separate two points with disjoint open nbds. If the sequence converges to both, then the sequence has to have finite number of elements outside of each nbd and infinite number of elements in each nbd which is a contradiction?

By the way you can simply do the proof by showing normed spaces are haussdorf (which should be easy using triangle inequality and concept of distance coming from both linearity and the norm) -and first countability if dextercioby is right-

 

[Fredrik]

I haven't even looked up what first countable means, but I don't think I need to. It's clear that the Hausdorff condition is sufficient, because $x_n\to x$ means that every open set that contains x also contains all but a finite number of terms of the sequence. If $x_n\to x$ and $x_n\to y\neq x$ in a Hausdorff space, then there are disjoint open sets U and V such that $x\in U$ and $y\in V$, and both contain all but a finite number of terms. This is a contradiction, because $x_n\to x$ implies that V can only contain a finite number of terms, and $x_n\to y$ implies that V must contain infinitely many.