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## Homework Statement

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.

## Homework Equations

The definition of convergence.

## 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) | < ε3. | (a

_{n}+ b

_{n}) - (A + B) | = | (a

_{n}- A) + (b

_{n}- B) | < ε

4. Since by hypothesis |a

_{n}- A| < ε/2, then

| (a

| (b

if n ≥ N

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

_{n}- B) | < ε/2if 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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