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.