Tail diameter of a sequence ($D_N={\sup }_{m,n\ge N}d(x_m,x_n)$)

The tail diameter of a sequence in a metric space is $D_N=\sup \{d(x_m,x_n):m,n\ge N\}$, allowing $+\infty$. The bounds are nonincreasing as $N$ grows because fewer pairs remain. A sequence is a Cauchy sequence exactly when $D_N\to 0$. The supremum need not be attained: for $q_n=1-2^{-n}$, $D_N=2^{-N}$ even though no two finite terms achieve that difference. For $(-1{)}^n$, $D_N=2$ for every $N$. For the harmonic numbers, $D_N=+\infty$ for every $N$, because each remaining tail is unbounded.