Problem: Show $ \frac{n+1}{4n^2 + 3} $ is a Cauchy sequence ( Using Cauchy Criterion only ).
Answer:
We'll show $ \forall n \in \mathbb{N}. a_{n+1} < a_n $
So given $ \epsilon >0 $ we'll chose $ N_{\epsilon} = \lceil \frac{1}{2\epsilon} \rceil $ and then we indeed showed that the given sequence is Cauchy sequence.
My question: When I attempted myself, naively I tried by deriving something useful and easy to see from $ a_{n+k} - a_{n} $ and then go ahead writing the proof more formally. But I got stuck in the middle because I'd get to an expression which is ugly and monstrous and it got me stuck. After looking at the answer, I wouldn't have thought to do as they did.
So I want to ask, how would you show the sequence is Cauchy sequence? Is there better alternative to the one purposed in the answer above?