I am trying to verify that $d_{\infty}((x_1,x_2),(y_1,y_2))= \max\{|x_1-y_1|,\,|x_2-y_2| \,\, \forall (x_1,x_2),(y_1,y_2)\in \mathbb{R^2}$ defines a metric. I have verified that $d_\infty$ satisfies the other conditions of a metric, apart from 'the triangle inequality'.
The following is what I did:
Let $(x_1,x_2),(y_1,y_2),(z_1,z_2) \in \mathbb{R^2}$.
Observe, the triangle inequality for real numbers, \begin{equation} |x_i-y_i| \leq |x_i-z_i|+|z_i-y_i|, \text{ where } i=1,2.\tag{$1$} \end{equation}
Then
\begin{align}
d_\infty((x_1,x_2),(y_1,y_2)) &:= \max\{|x_1-y_1|,|x_2-y_2|\} \\
&=\max\{|x_1-y_1 +z_1 -z_1 |,\,|x_2-y_2 +z_2-z_2|\} \\
&\leq \max\{|x_1-z_1 |+|z_1-y_1|,\,|x_2-z_2|+|z_2-y_2|\},
\end{align}
using $(1)$.
My question is whether $\max\{a+b,\,c+d\} \leq \max\{a,\,c\}+\max\{b,\,d\}$ for any $a,b,c,d \in \mathbb{R}$ is true, so that i can write:
\begin{align} &\max\{|x_1-z_1 |+|z_1-y_1|,\,|x_2-z_2|+|z_2-y_2|\} \leq \max\{|x_1-z_1 |, \, |x_2-z_2| \} + \max\{|z_1-y_1|, \,|z_2-y_2| \} \\ &:= d_\infty((x_1,x_2),(z_1,z_2))+d\infty((z_1,z_2),(y_1,y_2))? \end{align}
Intuitively it makes sense that $\max\{a+b,\,c+d\} \leq \max\{a,\,c\}+\max\{b,\,d\}$ is true, but would I need to prove sometheing like this or is it a known result?