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.

If we are given that the sequence sum of two sequences of positive real numbers converges to zero, does that mean that each sequence converges to zero? (by the squeeze theorem)

share|improve this question
3  
For sum reason this question was very hard to parse –  user58512 Feb 14 '13 at 17:55
    
How can the sum of a sequence of positive numbers converge to zero? –  Ross Millikan Feb 14 '13 at 18:05

2 Answers 2

Let $x_n$ and $y_n$ be your sequences.

The answer is yes since $$ 0\leq x_n\leq x_n+y_n\quad\mbox{and}\quad 0\leq y_n\leq x_n+y_n $$ and then you can apply the squeeze theorem.

share|improve this answer
    
What's going on today? Why the downvote? –  1015 Feb 14 '13 at 17:55
    
Why the downvote? –  Hagen von Eitzen Feb 14 '13 at 17:57
    
+1 This is a great answer and great reasoning. –  anon271828 Feb 14 '13 at 18:00
    
@anon271828 Thanks! But that's just the very beginning of a reasoning. –  1015 Feb 14 '13 at 18:02

If the sequence $a_n+b_n$ converges to zero and $a_n, b_n$ are positive then $a_n$ (and by symmetry $b_n$) converges to zero because it's bounded below by 0 and above by some sequence that tends towards zero.

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.