Possible metric space

Question

$d_m$ is defined on $\Bbb R^2$ as such:

$d_m(x,y) = max \lbrace|x_1 - y_1| , |x_2 -y_2| \rbrace $ where $x=(x_1,x_2) , y = (y_1 ,y_2)$

Which I have the task of proving whether or not the above s a matric.

I am having some difficultly with one of the properties of a metric, namely

$d(x,y) \leq d(x,z) + d(z,y)$ the famous triangle inequality.

I have a idea on how this could be proved

I thought of using the elementary properties of the modulus function i.e.:

for $i =1 , 2 $

$|x_i - y_i| = |x_i -z_i +z_i -y_i|\\ \leq |x_i -z_i|+|z_i -y_i|$

However , I am unsure how to show that this implies that

$d_m(x,y) \leq d_m(x,z) +d_m(z,y)$

I apologise for my horrible formatting

Answer 1

You are on the right track. Note that (using the inequality you already derived in my first inequality) $$ d(x,y) = \max\{|x_1-y_1|,|x_2-y_2|\} \le \max\{|x_1 - z_1| + |z_1-y_1|,|x_2-z_2| + |z_2-y_2|\} \le \max\{|x_1 - z_1|,|x_2-z_2|\} + \max\{|z_1-y_1|,|z_2-y_2|\} = d(x,z) + d(z,y). $$