Does $\lim\limits_{m\rightarrow\infty}\sum\limits_{k=m}^\infty d(a_k,a_{k+1})$ equal to zero for any sequence $(a_n)\subset M$

Question

Let $(M,d)$ be a metric space and $(a_n)\subset M$ a sequence. Is it always true that if $r_m=\sum\limits_{k=m}^\infty d(a_k,a_{k+1})$ we have $\lim\limits_{m\rightarrow\infty}r_m=0$?

$$r_m=\sum\limits_{k=0}^\infty d(a_k,a_{k+1})-\sum\limits_{k=0}^{m-1}d(a_k,a_{k+1})\xrightarrow[m\rightarrow\infty]{}\sum\limits_{k=0}^\infty d(a_k,a_{k+1})-\sum\limits_{k=0}^{\infty}d(a_k,a_{k+1})=0$$

Is this always correct or does the sequence have to satisfy $\sum\limits_{k=0}^\infty d(a_k,a_{k+1})<\infty$?

Answer 1

If $M$ is real line with usual metric and $a_n=1+\frac 1 2+\cdots+\frac 1 n$ then $r_m=\infty$ for all $m$. If you assume that $\sum d(a_n,a_{n+1})<\infty$ then $r_m \to 0$ as $m \to \infty$.