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I'm reading a proof that has this inequality which I figure follows somehow from the triangle ineq. but I can't figure it out.

I can provide more details if needed but I don't want to write out the whole proof, am I missing something simple here?

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  • $\begingroup$ Add $d(x,Tx)+d(y,Ty)$ on both sides, and then use the triangle inequality twice, to collapse the path $x \to Tx \to Ty \to y$ into $x \to y$. $\endgroup$ – Henning Makholm Nov 16 '15 at 12:22
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This is another way to write$$d(x,y)\leq d(x,Tx)+d(Tx,Ty)+d(Ty,y),$$which is simply the triangle inequality applied twice.

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