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.