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In any metric space $(M,D)$, prove that $D(a_1,a_n)\leq D(a_1,a_2)+D(a_2,a_3)+\cdots+D(a_{n-1},a_n)$ for any $a_1,a_2,a_3,\ldots,a_n \in M$

how to we approach this problem is we prove by Mathematical Induction

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    $\begingroup$ Just iterate the triangle inequality. $\endgroup$ – Francesco Polizzi Sep 6 '17 at 15:15
  • $\begingroup$ @FrancescoPolizzi..sorry i did't get can you give me one more step $\endgroup$ – Inverse Problem Sep 6 '17 at 15:18
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For $n=2$, it is trivial.

If it holds for a certain $n$, then\begin{align}D(a_1,a_{n+1})&\leqslant D(a_1,a_n)+D(a_n,a_{n+1})\\&\leqslant D(a_1,a_2)+D(a_2,a_3)+\cdots+D(a_{n-1},a_n)+D(a_n,a_{n+1}).\end{align}

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