(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose {a_{n}}_{n=1}^{∞}and {b_{n}}_{n=1}^{∞}are sequences such that {a_{n}}_{n=1}^{∞}and {a_{n}+ b_{n}}_{n=1}^{∞}converge.

Prove that {b_{n}}_{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 {a_{n}}_{n=1}^{∞}converges to A (by hypothesis).

Then for ε/2 > 0 there is a positive integer N_{1}such that if n ≥ N_{1}, then |a_{n}- A| < ε/2.

2. Assume that {a_{n}+ b_{n}}_{n=1}^{∞}converges to A + B (by hypothesis).

Then for ε > 0 there is a positive integer N = max{N_{1}, N_{2}} such that if n ≥ N, then | (a_{n}+ b_{n}) - (A + B) | < ε

3. | (a_{n}+ b_{n}) - (A + B) | = | (a_{n}- A) + (b_{n}- B) | < ε

4. Since by hypothesis |a_{n}- A| < ε/2, then

| (a_{n}- A) - (a_{n}- A) + (b_{n}- B) | < ε - ε/2

| (b_{n}- B) | < ε/2

if n ≥ N_{2}for some positive integer N_{2}.

5. But this is the definition of convergence, therefore {b_{n}}_{n=1}^{∞}converges (to B). □

Thanks.

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

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