How Can I Solve These Sequence Problems Quickly?

[rahl__]

i have a few problems with sequences
1. show, that if:
$$\lim_{n\to\infty}a_{n}=L$$
than sequence:
$$b_{n}=\frac{a_{1}+...+a_{n}}{n}$$
is convergent to L

2. show that the sequence$$a_{n}$$ is monotone, bounded and find out its limit, if:
$$a_{1}=2$$
$$a_{n+1}=\frac{a_{n}+4}{2}$$

3. show that if the sequence $$a_{n}$$ satysfies cauchy's condition than it is convergent.

4. show that there is an inequility :
$$|\sum_{k=1}^{n}a_{k}b_{k}|\leq\sqrt{\sum_{k=1}^{n}a_{k}^{2}}\sqrt{\sum_{k=1}^{n}b_{k}^{2}}$$

5. find the limit of such sequence:
$$a_{n}=(\frac{n+1}{n})^{3n^{2}}$$

6. find the limit of such sequence:
$$a_{n}=(\frac{n^{2}+4}{n^{2}+3})^{2n}$$

7. find the limit of such sequence
$$a_{n}=-n^{6}+3n^{5}+7$$

8. find the limit of such sequence
$$a_{n}=\sqrt[n]{n!}$$

9. find the limit of such sequence
$$a_{n}=1+2^{n}-3^{n}$$

10. $$a_{n}$$ is a sequence including all rational numbers. show that for each real number M you can find a subsequence of this sequence that is convergent to M

11. $$a_{n}$$ is a squence, that has a subsequence convergent to $$\infty$$ and a subsequence convergent to -$$\infty$$. show that, if $$\lim_{n\to\infty}(a_{n}-a_{n-1})=0$$, than for each real number M there is a subsequence convergent to M.

thanks in advance and sorry for the length of this post, but i really need this answers as soon as possible

 

[VietDao29]

thanks in advance and sorry for the length of this post, but i really need this answers as soon as possible

Whoops, sorry, but we are not giving out COMPLETE SOLUTIONS to those who do not even bother to try to find a way to tackle the problem(s). Why must we help him if he shows no interest in finding the solutions on this own? And to remind you, it's your own problems, not ours...
You may want to read the https://www.physicsforums.com/showthread.php?t=28.
Now, may you just show us your works, what have you done to go about tackling the problems? Or at least, some of your thoughts about the problems. And we may help you.

 

[rahl__]

you took me wrong, these problems were not my homework exercises. I had an exam last saturday from sequences and multitude theory [?] and from a list of about 100 exercises these 11 were the ones that i had some problems when trying to solve. i thought that this exam will be in 2 weeks time, but it turned out to be on previous saturday so i had very little time to do all those exercises and that's why i have posted just bare examples without my thoughts regarding the possible sollution, its not like I am that lazy or sth.
sorry for creating such confusion

 

[siddharth]

I'm sorry rahl, but the https://www.physicsforums.com/showthread.php?t=5374", which you agreed to, says:

On helping with questions: Any and all assistance given to homework assignments or textbook style exercises should be given only after the questioner has shown some effort in solving the problem. If no attempt is made then the questioner should be asked to provide one before any assistance is given. Under no circumstances should complete solutions be provided to a questioner, whether or not an attempt has been made.

So, when you show your efforts, thoughts or any ideas on the problems, people like Vietdao29 and others will gladly assist you.