## Convergent sequences: if {an} converges and {an + bn} converges, prove {bn} converges

1. The problem statement, all variables and given/known data

Suppose {an}n=1 and {bn}n=1 are sequences such that {an}n=1 and {an + bn }n=1 converge.

Prove that {bn}n=1 converges.

2. Relevant equations

The definition of convergence.

3. The attempt at a solution

I am pretty new to mathematics that requires proof, so excuse me if I do something really stupid... but basically, is this a sufficient proof?

1. Assume {an}n=1 converges to A (by hypothesis).
Then for ε/2 > 0 there is a positive integer N1 such that if n ≥ N1, then |an - A| < ε/2.
2. Assume that {an + bn }n=1 converges to A + B (by hypothesis).
Then for ε > 0 there is a positive integer N = max{N1, N2} such that if n ≥ N, then | (an + bn) - (A + B) | < ε
3. | (an + bn) - (A + B) | = | (an - A) + (bn - B) | < ε

4. Since by hypothesis |an - A| < ε/2, then
| (an - A) - (an - A) + (bn - B) | < ε - ε/2

| (bn - B) | < ε/2

if n ≥ N2 for some positive integer N2.
5. But this is the definition of convergence, therefore {bn}n=1 converges (to B).

Thanks.
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 Blog Entries: 1 Recognitions: Homework Help You never actually say what N2 is. Also, if a+b
 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus There's a problem in step 4 of the attempt. It doesn't follow from | (an - A) + (bn - B) | < ε that | (an - A) - (an - A) + (bn - B) | < ε - ε/2 It's like saying |1-1.9|=0.9 < 1 so |1-1-1.9| < 1-1=0.

Recognitions:
Homework Help

## Convergent sequences: if {an} converges and {an + bn} converges, prove {bn} converges

This looks okay, why not look at the algebra of limits? if $$a_{n}+b_{n}\rightarrow b}$$ and $$a_{n}\rightarrow a$$ then the sequence $$b_{n}=a_{n}+b_{n}-a_{n}\rightarrow b-a$$
 Thanks everyone. I'm clearly missing something in the understanding of this material, so I'll take what you've said this weekend and dig through the book and see if I can spot the misunderstanding.