Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

EDIT: Apparently I translated the terminology wrong, but partial limit means limit of a subsequence.

If $a_{n}$ has 'a' partial limits and $b_{n}$ has 'b', can $a_{n}+b_{n}$ have more than $ab$ partial limits? ('infinity' also counting as a limit)

This is a homework question. I'm thinking the statement is false, but I have no idea where to begin! Could someone give me a hint?

Thanks!

share|improve this question
1  
"partial limit"? –  David Mitra Nov 17 '11 at 22:29
    
Maybe I translated it wrong? It is a limit of a sub-sequence of $a_{n}$, e.g. '1' and '-1' are partial limits of 1,-1,1,-1,... –  rowe Nov 17 '11 at 22:30

2 Answers 2

up vote 4 down vote accepted

I take it by "partial limit" you mean limit of a subsequence. It is possible for $a_n$ and $b_n$ to have zero partial limits each, while $a_n+b_n$ has a limit. E.g., $a_n=n$, $b_n=-n$.

share|improve this answer
    
I believe the question refers to limits in the broad sense, that is infinity is a valid limit... I've edited it to say as much! –  rowe Nov 17 '11 at 22:32
    
Change $a_n$ to two different sequences which go to infty... $a_{2n}=2n$ and $a_{2n+1}=2n-2$ for example... $b_n=-n$ is unchanged.... –  N. S. Nov 17 '11 at 22:36
    
I'm not sure but the answer might change if your sequences are bounded ..... –  N. S. Nov 17 '11 at 22:37
    
N.S.: In your example $a_{n}$ has one limit (infinity), $b_{n}$ has one limit (negative infinity), and $a_{n}+b_{n}$ has exactly $1*1=1$ limits (infinity), so I suspect this doesn't work. –  rowe Nov 17 '11 at 22:38
1  
Interestingly enough, it does indeed work for a bounded sequence. –  Phira Nov 17 '11 at 23:11

The claim is true if the sequences involved are bounded.

Let $a_{n_k}+b_{n_k}$ be a convergent subsequence of the sum sequence.

Since the sequence $a_{n_k}$ is bounded, it has a converging subsequence $a_{n_{k_i}}$ that converges to some limit point of $a_n$.

This means that $b_{n_{k_i}}$ is the difference of two converging sequences, so it converges as well and its limit is a limit point of $b_n$.

Therefore, the limit of the convergent sequence $a_{n_k}+b_{n_k}$ which is of course the limit of its subsequence is the sum of a limit point of $a_n$ and a limit point of $b_n$.

There are only $a\cdot b$ such sums, therefore the total number is at most $a\cdot b$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.