Triangle inequality for $\ell^1$ metric on $\mathbb{R}^n$.

Question

How can I show that the distance satisfies the triangle inequality?

Where the distance is given by: $$d(p,q) = |p_1-q_1| + \cdots + |p_n-q_n|$$

Triangle Inequality: $\forall x,y,z\Bigl( d(x,z) \leq d(x,y) + d(y,z)\Bigr)$.

Answer 1

To prove the triangle inequality for this metric it is sufficient to note that each of $|x_i-z_i| \le |x_i-y_i|+|y_i-z_i|$ by the simple 1-dimensional triangle inequality.

Answer 2

Hint: Use the triangle inequality for the absolute value on $\mathbb{R}$.