= Tail diameter of a sequence
{title2=$D_N=\sup_{m,n\geq N}d(x_m,x_n)$}
The tail diameter of a <sequence> in a <metric space> is $D_N=\sup\{d(x_m,x_n):m,n\geq N\}$, allowing $+\infty$. The bounds are nonincreasing as $N$ grows because fewer pairs remain. A <sequence> is a <Cauchy sequence> exactly when $D_N\to0$. 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.
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