# Converging sequences

1. Jul 27, 2008

### Nan1teZ

1. The problem statement, all variables and given/known data

Prove or give a counterexample: If {an} and {an + bn} are convergent sequences, then {bn} 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?

2. Jul 27, 2008

### Dick

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

3. Jul 27, 2008

### 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?

4. Jul 28, 2008

### konthelion

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

5. Jul 28, 2008

### Dick

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