How is the triangle inequality being used here?

Question

I'm reading the Banach fixed point theorem (https://en.wikipedia.org/wiki/Banach_fixed-point_theorem) and at one point, this inequality is used: https://puu.sh/AS4uY/78caef5bc6.png

However, I'm not seeing how it's being used here. Any insight on it would be really helpful

Answer 1

The triangle inequality is being used several times (or by induction on $m-n$ depending on how precisely you want to present the proof)

First step: $$d(x_m,x_n)\leq d(x_m,x_{m-1})+d(x_{m-1},x_n)$$ This is just the triangle inequality applied to $x_m,x_{m-1},x_n$

Second: $$d(x_{m-1},x_n)\leq d(x_{m-1},x_{m-2})+d(x_{m-2},x_n)$$ which is triangle inequality for $x_{m-1},x_{m-2},x_n$. And now use transitivity of $\leq$ with this inequality and the one from the previous step to obtain $$d(x_m,x_n)\leq d(x_m,x_{m-1}) + d(x_{m-1},x_{m-2})+d(x_{m-2},x_n)$$

Third: Apply triangle inequality to $x_{m-2},x_{m-3},x_n$ and transitivity with the previous inequality.

...

Eventually the last term in the right of the inequality becomes $d(x_{n+1},x_n)$, and then you stop.