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

Nan1teZ
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Homework Statement



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?
 
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Yes, and M=L+N, right? You can certainly prove that. It's the correct approach.
 
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?
 
Hint: \lim_{n \to \infty} \left\{ a_{n} + b_{n} \right\} = \lim_{n \to \infty} a_{n} + \lim_{n \to \infty} b_{n}
 
And bn=(an+bn)+(-an).
 
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