Proving Converging Sequences: {an}, {an + bn}, {bn}

[Nan1teZ]

Homework Statement

Prove or give a counterexample: If {a_n} and {a_n + b_n} are convergent sequences, then {b_n} is a convergent sequence.


2. The attempt at a solution

Ok I couldn't think of any counterexamples, so I tried to prove it using delta epsilon definitions:

|an - L| < E
|an + bn - M| < E
want to show: |bn - N| < E

Is this the right approach?

 

[Dick]

Yes, and M=L+N, right? You can certainly prove that. It's the correct approach.

 

[Nan1teZ]

yeah I got the N = M-L part. But then after that I go in circles trying to show it is < Epsilon. =[

What's the little trick?

 

[konthelion]

Hint: $$\lim_{n \to \infty} \left\{ a_{n} + b_{n} \right\} = \lim_{n \to \infty} a_{n} + \lim_{n \to \infty} b_{n}$$

 

[Dick]

And bn=(an+bn)+(-an).