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.