There is definition of Cauchy sequence in the book of Introduction to Calculus and classical analysis by Omar hijab and that is :
$\forall n,m\in \Bbb N \space e_n \ge 0 ,e_n\to 0 \space ,|a_{m+n}- a_n|\lt e_n $
$e_n$ is a error sequence for cauchy sequence .and there is a general definition which many books including Rudin use which is :
$\forall\epsilon\gt0 \space\space\exists N\in\Bbb N \space \space s.t \space \space\forall m,n\ge N \space\space |a_n-a_m|\lt\epsilon$
I want to prove that these definitions are equivalent.I don't have any idea how to prove it !
1) => 2) is quite obvious: $(e_n) \rightarrow 0 $ => For any $ \epsilon >0 $, there is a N such as: $ n \geq N $ => $ e_n \leq \epsilon $
=> $\forall\epsilon\gt0 \space\space\exists N\in\Bbb N \space \space s.t \space \space \forall m\in\Bbb N ;\forall n\ge N \space\space |a_{n+m}-a_n|\lt\epsilon $
That is the same as: $\forall\epsilon\gt0 \space\space\exists N\in\Bbb N \space \space s.t \space \space\forall m,n\ge N \space\space |a_n-a_m|\lt\epsilon$
2) => 1) : Starting with : $\forall\epsilon\gt0 \space\space\exists N\in\Bbb N \space \space s.t \space \space\forall m,n\ge N \space\space |a_n-a_m|\lt\epsilon$
Since one can switch the role of n and m, we can assume that m> n . Let's write m=n+p
2) <=> $ \forall\epsilon\gt0 \space\space\exists N\in\Bbb N \space \space s.t \space \space\forall p\in\Bbb N,n\ge N \space\space |a_n-a_{n+p}|\lt\epsilon $
Here the last condition allows you to define : $\epsilon_n = sup(|a_n-a_{n+p}|;p\in\Bbb N)$ since this sequence of p is always bounded.
You then have by definition : $ n \leq N : |a_n-a_{n+p}| \leq \epsilon_n $ ; $ n > N : |a_n-a_{n+p}| \leq \epsilon_n \leq \epsilon $
Hence this sequence can be dominated by an other sequence $(\epsilon_n)$ that $\rightarrow 0$, which is your first definition.