Proving Cauchy Sequence Subsequences

[clueles]

I need help on trying to prove

that every subsequence of a cauchy sequence is a cauchy sequence

 

[Jimmy Snyder]

If [itex]\{x_{n_{k}}\}[/itex] is a subsequence of {[tex]x_{n}[/tex]}, then for all [tex]k[/tex], [tex]n_{k} \ge k[/tex]. So the '[tex]N[/tex]' that works for {[tex]x_{n}[/tex]} also works for {[tex]x_{n_{k}}[/tex]}

 

[clueles]

is that how you would write a formal proof or would you the for any epsilon>0 there exists an n in i such that abs(sm-sn)<epsilon

 

[Jimmy Snyder]

The answer to that depends on the audience. What is your situation?

 

[clueles]

college level for a real analysis class

 

[Jimmy Snyder]

Please note the small change I made in the proof. At the undergraduate level, probably the full [tex]\epsilon[/tex] treatment would be best. Are you up to it?

 

[clueles]

Yes we are up to that part but I'm not sure how to even start it. Do we need to say for all [tex]\epsilon[/tex]> there exists a n,m /geqN such that abs(Snk-Smk)<E?

 

[Galileo]

I'd start with the definition of a subsequence.
Also, always write down the hypothesis (what you know) and what you need to prove:

Let [itex]\{x_{n_k}\}[/itex] be a subsequence of [itex]\{x_n\}[/itex].

You are given that for every [itex]\epsilon>0[/itex], there exists an integer N>0, such that:

[tex]m,n > N \Rightarrow |x_n-x_m|<\epsilon[/itex]

You need to prove that for every [itex]\epsilon>0[/itex] there exists an integer N>0, such that:

[tex]m_k,n_k > N \Rightarrow |x_{n_k}-x_{m_k}|<\epsilon[/tex]

All I've done is written down the problem in different terms, but the answer is practically staring you in the face (hopefully).

 

[Jimmy Snyder]

[itex]\{x_{n_k}\}[/itex]

I'm just starting to use tex and I only know what I see in other people's examples. Your stuff is nice and I learned a lot from it. Do you know of a Windows utility that will take a tex source file and create a pdf file from it?

 

[Galileo]

I'm just starting to use tex and I only know what I see in other people's examples. Your stuff is nice and I learned a lot from it. Do you know of a Windows utility that will take a tex source file and create a pdf file from it?

Yes, WinEdt will do that for you. I think you have to register it or something. I use it often to create pdf's.
The only advantage of not registering is annoying popup reminders every 15 seconds or so. I actually use a different editor and use only WinEdt for creating the pdf's.

 

[Jimmy Snyder]

Thanks, I will look into winedt. In the meantime, I found a fairly large package called miktex which has a utility with the unlikely name of pdftex that turns tex into pdf.

 

[Data]

TeXnic Center is the IDE that I use in general. It has full TeX-to-pdf capability and is built to work with Miktex as well (it requires a TeX install to work, of course). Free too~

It's a little hard to find these days though (although it is quite popular).

If you want, I can probably find somewhere to host it for a few days. It's only about 5mb.

 

[matt grime]

You don't, as far as I can recall (linux for the last 3 years), need any of those IDE's or such. To use latex on MS you need Miktex installed. That comes with the program pdflatex, I think, that creates the pdf for you. It means learning to love the command line though. just type

pdflatex foo.tex

from inside the directory where foo.tex is stored and it outputs a pdf file for you. The installation also comes with dvips which creates a post script file from the pdf. There are other format generators too, including latex2html, though I've no idea if anyone acutally uses this anymore.