Cauchy's criterion (for sequences) says that a sequence $(a_n)$ converges if and only if for every $\epsilon >0$ there exists a natural number $N$ such that for all natural numbers $m,n>N$ we have $|a_m-a_n|<\epsilon$.However, "Cauchy's criterion without an absolute value" is also true. Namely, a sequence $(a_n)$ converges if and only if for every $\epsilon >0$ there exists a natural number $N$ such that for all natural numbers $m,n>N$ we have $a_m-a_n<\epsilon$. The "only if" part is true from Cauchy's criterion and the fact that $x\leq |x|$ for every $x\in\mathbb{R}$. The "if" part is true from Cauchy's criterion and the fact that $|a_m-a_n|\in\{a_m-a_n,a_n-a_m\}$.
"Cauchy's criterion without an absolute value" makes things easier in many cases, because often when one proves that a sequence is Cauchy, one needs to use the triangle inequality.Why then is "Cauchy's criterion without an absolute value" not the standard form of Cauchy's criterion?