# 1.1 Convergent sequences, cauchy sequences, and complete spaces  (Page 2/2)

 Page 2 / 2

Proof: Assume that a sequence ${x}_{n}\to x$ in $\left(X,d\right)$ . Fix $ϵ>0$ . Since $\left\{{x}_{n}\right\}$ is convergent, there must exist an ${n}_{0}\in {\mathbb{Z}}^{+}$ such that $d\left({x}_{n},x\right)<ϵ/2$ for all $n\ge {n}_{0}$ . Now, pick $j,k\ge {n}_{0}$ . Then, using the triangle inequality, we have $d\left({x}_{j},{x}_{k}\right)\le d\left({x}_{j},x\right)+d\left(x,{x}_{k}\right)<ϵ/2+ϵ/2$ (since both $j$ and $k$ are greater or equal to ${n}_{0}$ ) and so $d\left({x}_{j},{x}_{k}\right)<ϵ$ . Therefore, the sequence $\left\{{x}_{n}\right\}$ is Cauchy.

One may wonder if the opposite is true: is every Cauchy sequence a convergent sequence?

Example 3 Focus on the metric space $\left(X,{d}_{0}\right)$ with $X=\left(0,1\right]=\left\{x\in \mathbb{R}:0 . Now consider the sequence ${x}_{n}=1/n$ in $X$ . This is a Cauchy sequence: one can show this by picking ${n}_{0}\left(ϵ\right)>⌈2/ϵ⌉$ and using the triangle inequality to get

${d}_{0}\left({x}_{j},{x}_{k}\right)\le {d}_{0}\left({x}_{j},0\right)+{d}_{0}\left({x}_{k},0\right)=|{x}_{j}|+|{x}_{k}|=\frac{1}{j}+\frac{1}{k}\le \frac{1}{{n}_{0}}+\frac{1}{{n}_{0}}=\frac{2}{{n}_{0}}<\frac{2}{⌈2/ϵ⌉}\le \frac{2}{2/ϵ}=ϵ.$

However, this is not a convergent sequence: we've shown earlier that ${x}_{n}\to 0$ . Since $0\notin \left(0,1\right]$ and a sequence has a unique limit, then there is no $x\in \left(0,1\right]$ such that ${x}_{n}\to x$ .

## Complete metric spaces

Whether Cauchy sequences converge or not underlies the concept of completeness of a space.

Definition 4 A complete metric space is a metric space in which all Cauchy sequences are convergent sequences.

Example 5 The metric space $\left(X,{d}_{0}\right)$ with $X=\left(0,1\right]$ from earlier is not complete, since we found a Cauchy sequence that converges to a point outside of $X$ . We can make it complete by adding the convergence point, i.e., if ${X}^{\text{'}}=\left[0,1\right]$ then $\left({X}^{\text{'}},{d}_{0}\right)$ is a complete metric space.

Example 6 Let $C\left[T\right]$ denote the space of all continuous functions with support $T$ . If we pick the metric ${d}_{2}\left(x,y\right)={\left({\int }_{T},{|x\left(t\right)-y\left(t\right)|}^{2},d,t\right)}^{1/2}$ , then $\left(X,{d}_{2}\right)$ is a metric space; however, it is not a complete metric space.

To show that the space is not complete, all we have to do is find a Cauchy sequence of signals within $C\left[T\right]$ that does not converge to any signal in $C\left[T\right]$ . For simplicity, fix $T=\left[-1,1\right]$ . Fortuitously, we find the sequence illustrated in [link] , which can be written as

${x}_{n}\left(t\right)=\left\{\begin{array}{cc}-1\hfill & \mathrm{if}\phantom{\rule{3.33333pt}{0ex}}t<-1/n,\hfill \\ 1\hfill & \mathrm{if}\phantom{\rule{3.33333pt}{0ex}}t>1/n,\hfill \\ nt\hfill & \mathrm{if}\phantom{\rule{3.33333pt}{0ex}}-1/n\le t\le 1/n.\hfill \end{array}\right)$

Since all these functions are continuous and defined over $T$ , then $\left\{{x}_{n}\right\}$ is a sequence in $C\left[T\right]$ . We can show that $\left\{{x}_{n}\right\}$ is a Cauchy sequence: let ${n}_{0}$ be an integer and pick $j\ge k\ge {n}_{0}$ . Then,

$\begin{array}{cc}\hfill {d}_{2}\left({x}_{j},{x}_{k}\right)& ={\left({\int }_{-1}^{1},{|{x}_{k}\left(t\right)-{x}_{j}\left(t\right)|}^{2},d,t\right)}^{1/2}={\left({\int }_{-1/j}^{1/j},{|{x}_{k}\left(t\right)-{x}_{j}\left(t\right)|}^{2},d,t\right)}^{1/2}\le {\left({\int }_{-1/j}^{1/j},{1}^{2},d,t\right)}^{1/2}\hfill \\ & \le {\left(2/j\right)}^{1/2}\le \sqrt{2/{n}_{0}}.\hfill \end{array}$

So for a given $ϵ>0$ , by picking ${n}_{0}$ such that $\sqrt{2/{n}_{0}}<ϵ$ (say, for example, ${n}_{0}>⌈2/{ϵ}^{2}⌉$ ), we will have that ${d}_{2}\left({x}_{j},{x}_{k}\right)<ϵ$ for all $j,k>{n}_{0}$ ; thus, the sequence is Cauchy. Now, we must show that the sequence does not converge within $C\left[T\right]$ : we will find a point ${x}^{*}\notin C\left[T\right]$ such that for ${X}^{\text{'}}=C\left[T\right]\cup \left\{{x}^{*}\right\}$ the sequence ${x}_{n}\to {x}^{*}$ in $\left({X}^{\text{'}},{d}_{2}\right)$ . By inspecting the sequence of signals, we venture the guess

${x}^{*}\left(t\right)=\left\{\begin{array}{cc}-1\hfill & \mathrm{if}\phantom{\rule{3.33333pt}{0ex}}t<0,\hfill \\ 1\hfill & \mathrm{if}\phantom{\rule{3.33333pt}{0ex}}t>0,\hfill \\ 0\hfill & \mathrm{if}\phantom{\rule{3.33333pt}{0ex}}t=0.\hfill \end{array}\right),$

For this signal, we will have

$\begin{array}{cc}\hfill {d}_{2}\left({x}_{n},{x}^{*}\right)& ={\left({\int }_{-1}^{1},{|{x}_{n}\left(t\right)-{x}^{*}\left(t\right)|}^{2},d,t\right)}^{1/2}={\left({\int }_{-1}^{0},|,{x}_{n},\left(t\right),-,{x}^{*},{\left(t\right)|}^{2},d,t,+,{\int }_{0}^{1},{|{x}_{n}\left(t\right)-{x}^{*}\left(t\right)|}^{2},d,t\right)}^{1/2}\hfill \\ & ={\left({\int }_{-1/n}^{0},{|nt-\left(-1\right)|}^{2},d,t,+,{\int }_{0}^{1/n},{|nt-1|}^{2},d,t\right)}^{1/2}\hfill \\ & ={\left({\int }_{-1/n}^{0},{|nt+1|}^{2},d,t,+,{\int }_{0}^{1/n},{|1-nt|}^{2},d,t\right)}^{1/2}\hfill \\ & ={\left({\int }_{0}^{1/n},{|1-nt|}^{2},d,t,+,{\int }_{0}^{1/n},{|1-nt|}^{2},d,t\right)}^{1/2}={\left(2,{\int }_{0}^{1/n},{\left(1-nt\right)}^{2},d,t\right)}^{1/2}\hfill \\ & ={\left(\frac{2}{3n}\right)}^{1/2}.\hfill \end{array}$

So if we select ${n}_{0}$ such that ${\left(\frac{2}{3{n}_{0}}\right)}^{1/2}<ϵ$ , i.e., ${n}_{0}>\frac{2}{3{ϵ}^{2}}$ , then we have that ${d}_{2}\left({x}_{n},{x}^{*}\right)<ϵ$ for $n>{n}_{0}$ , and so we have shown that ${x}_{n}\to {x}^{*}$ . Now, since a convergent sequence has a unique limit and ${x}^{*}\notin C\left[T\right]$ , then $\left\{{x}_{n}\right\}$ does not converge in $\left(C\left[T\right],{d}_{2}\right)$ and this is not a complete metric space.

The property of equivalence between Cauchy sequences and convergent sequences often compels us to define metric spaces that are complete by choosing the metric appropriate to the signal space. For example, by switching the distance metric to ${d}_{\infty }\left(x,y\right)={sup}_{t\in T}|x\left(t\right)-y\left(t\right)|$ , the metric space $\left(C\left[T\right],{d}_{\infty }\right)$ becomes complete.

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!