I am going to prove $a_n=\dfrac{n^2}{2n^2+3}$ is Cauchy.
This is what I did:
Let $\epsilon>0$.
$|a_n-a_m|=\dfrac{3}{4}.\dfrac{n^2+m^2}{n^2 m^2}<\dfrac{n^2+m^2}{n^2 m^2}=\dfrac{1}{m^2}+\dfrac{1}{n^2}$
$m,n$ are index of sequence elements.
My question is : Am I allowed to conclude $\dfrac{1}{m^2}+\dfrac{1}{n^2}<\dfrac{1}{m}+\dfrac{1}{n}$
I mean are $m,n$ always considered to be positive integers? so that as the last part of the proof can I state:
for the given $\epsilon>0$ we choose an $N\in\mathbb{N}$ s.t $N>\dfrac{2}{\epsilon}$.
Then for all $n,m\ge N$, We have
$|a_n-a_m|<\dfrac{1}{n^2}+\dfrac{1}{m^2}<\dfrac{1}{N}+\dfrac{1}{N}=\dfrac{2}{N}<\epsilon$
Does this sound correct to you? What do you recomment to prove the sequence above is Cauchy?