= Summable increments imply a Cauchy sequence
{title2=$\sum_{n=1}^\infty d(x_{n+1},x_n)<\infty\implies(x_n)\text{ is Cauchy}$}
If a <sequence> in a <metric space> has $\sum_{j=1}^\infty d(x_{j+1},x_j)<\infty$, the <triangle inequality> gives $d(x_m,x_n)\leq\sum_{j=n}^{m-1}d(x_{j+1},x_j)$ for $m>n$. The tail of this <convergent series> tends to zero and bounds every later pair, proving the <Cauchy sequence> condition. In particular, an increment bound $d(x_{j+1},x_j)\leq C r^j$ with $C>0$ and $0<r<1$ gives $d(x_m,x_n)\leq C r^n/(1-r)$ by the <geometric series>. Unlike checking a small latest step, this bounds all remaining movement.
This sufficient condition is not necessary. The real <sequence> $x_n=(-1)^n/n$ converges to zero and is therefore a <Cauchy sequence>, although $|x_{n+1}-x_n|=1/(n+1)+1/n$ has a divergent sum. Cancellation allows convergence despite infinite accumulated <absolute values> of the successive changes.
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