Correlation between the convergence of a sequence and subsequence

Hi guys I have 3 questions,

1-)(a_{n}) is a sequence that converges to L. How can I prove that any subsequence of (a_{n}) converges to L too?

2-)Where can I post questions of Linear Algebra? There isn't a section like that, maybe Linear Algebra belongs to a specific subject that I don't know?

3-) I am a first year Math student. I need advice from people who already finished University. How can I study? How can I study especially to my Analysis (Inifitesimal Calculus) course? The lecturer's notes aren't always very clear. And sometimes I feel helpless and I write here or ask people. I feel bad every time I get help because I feel that I have to be the one who solve the things. So I will be happy to get suggestions, how to study, or maybe there are good and clear books that someone recommend that would help me study without being dependent on the lecturer's note..

Anyway thanks in advance!

Re: Correlation between the convergence of a sequence and subsequence

Quote:

Originally Posted by

**davidciprut** Hi guys I have 3 questions,

1-)(a_{n}) is a sequence that converges to L. How can I prove that any subsequence of (a_{n}) converges to L too?

2-)Where can I post questions of Linear Algebra? There isn't a section like that, maybe Linear Algebra belongs to a specific subject that I don't know?

3-) I am a first year Math student. I need advice from people who already finished University. How can I study? How can I study especially to my Analysis (Inifitesimal Calculus) course? The lecturer's notes aren't always very clear. And sometimes I feel helpless and I write here or ask people. I feel bad every time I get help because I feel that I have to be the one who solve the things. So I will be happy to get suggestions, how to study, or maybe there are good and clear books that someone recommend that would help me study without being dependent on the lecturer's note..

Anyway thanks in advance!

Regarding 2 Advanced Algebra is a reasonable place to post. I wouldn't get too hung up about where to post your question.

Regarding 3, specifically regarding the line "I feel bad every time I get help because I feel that I have to be the one who solve the things." Are you serious? You feel badly because you are the one that has to solve things related to the coursework you are doing? Who do you think should be doing the solving and if it's not you how are you ever going to learn anything? Remember what you pay for getting questions answered here. If you want someone to do all your work for you go hire someone, but you had better hire them to take your tests as well.

If above was just a glitch in translating from Hebrew to English well ok then but if not you really need to revise your attitude a bit. Learning involves work. There is no way around that. Sorry.

regarding 1 - here is a quick proof

Re: Correlation between the convergence of a sequence and subsequence

First of all thanks for replying!

But For my third question you got me wrong, I do a lot of work, but sometimes despite the hard work there are questions that that I still have trouble solving it, and a lot of times the lecturers notes aren't very clear and the asistants in the university don't explain it well either, so in these kind of situation i just feel helpless ,but still if I get help i dont feel good with it because i didnt achieve it at the end despite the hard work. For it's my first year im still in the "getting used to" process to university, i just try to understand how to handle everything.. Anyway, thanks for the proof !

Re: Correlation between the convergence of a sequence and subsequence

Quote:

Originally Posted by

**davidciprut** 1-)(a_{n}) is a sequence that converges to L. How can I prove that any subsequence of (a_{n}) converges to L too?

Here is a very intuitive proof.

To say $\displaystyle (a_n)\to L$ means that almost all (all but a finite collection) of the terms of $\displaystyle (a_n)$ are *close to* $\displaystyle L$.

Therefore, the same must be true of any subsequence of the given sequence.