A sequence $x_i$ is Cauchy if for all $r>0$, there exists $n$ s.t. $i,j\geq n$ implies $d(x_i,x_j)<r$.
My question is, is it equivalent to define Cauchy as follows? $x_i$ is Cauchy if for all $r>0$, there exists $n$ such that $i>n$ implies $d(x_n,x_i)<r$.
If $x_n$ is Cauchy then it obviously satisfies my definition (just set $i=n$). If it satisfies my definition, then let's say we're given an $r$. By my definition, there exists $n$ s.t. $i\geq n$ implies $x_i\in B(x_n,r/2)$. So if $i,j\geq n$, then $x_i,x_j$ are both in this ball, so $d(x_i,x_j)<r$ by triangle inequality.
I ask because my version seems easier to prove. You only have to consider $(n,i)$ s.t. $i>n$ instead of all pairs $(i,j)$ with $i,j\geq n$. It's also nicer to visualize because I can imagine all $x_i$ s.t. $i\geq n$ contained in this nice ball.