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So I am trying to prove that the taxicab distance using the triangular inequality.
$d_1(p,q)=\|p-q\|_1=\sum_{i=1}^n|p_i-q_i|$

So I am trying to show that: $|d_1(a,b)−d_1(c,b)| \le d(a,c)$

which is: $\sum_{i=1}^n |a(i) - b(i)| - \sum_{i=1}^n |c(i) - b(i)| ≤ \sum_{i=1}^n |a(i) - c(i)|$

How would I combine the 2 sum series on the left?

Thank you.

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    |a(i) - b(i)| ≤ |a(i) - c(i)| + |c(i) - b(i)| then I can do sqrt((a-b)^2) ≤ sqrt((a-c)^2) + sqrt((c-b)^2) I expanded that out but I don't know what to do from there.2011-10-23

1 Answers 1

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It is enough to show that $d(a,b) \leq d(a,c)+ d(c,b),$ since then, we also have $d(a,c) - d(c,b) \leq d(a,b)$.

In other words, we aim to show that

$ d(a,b) = \sum_{i=1}^n |a_i - b_i| \leq \sum_{i=1}^n \left( |a_i - c_i| + |c_i - b_i|\right), $

and to do this, all we need to do is show $|x + y| \leq |x| + |y|$, where $x, y \in \mathbb{R}$ and $|\cdot|$ is the usual absolute value. I will leave it to you to fill in the details.

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    No, not a problem!2011-10-23