Analysis: Please check my proof on the arithmetic of CONVERGENT SEQUENCES

I am new to analysis and am having a lot of trouble with this proof. I am not sure if what I have written is valid. Please let me know. Thank you.

** Theorem:** If {an} converges to A and {an + bn} converges to A + B, then {bn} converges.

** Proof:**

Since {an + bn} converges to A + B, for every ε> 0 there exists an N such that for

n > N, |(an + bn) - (A + B)| < ε

=>

|(an - A) + (bn - B)| < ε

It is true that

|(an - A) + (bn - B)| ≤ |an - A| + |bn - B|

It is also true that

|(an - A) + (bn - B)| - |an - A| < ε for n > N

Thus it is true that

|(an - A) + (bn - B)| - |an - A| ≤ |an - A| + |bn - B| - |an - A| < ε

for n > N

Therefore

|an - A| + |bn - B| - |an - A| < ε for n > N

so

|bn - B| < ε for n > N

Since for ε there exists an N such that n > N => |bn - B| < ε, then {bn} converges.■

I used the same N for {bn} that I used for {an +bn} because if it works for

{an + bn} and {bn} is inside {an + bn}, then I reason it will work for {bn} too.