## Sum of two metrics is a metric

It is quite straightforward to prove or verify that the sum of two metrics (distance functions) is still a metric.

Suppose $d_1(x,y)$ and $d_2(x,y)$ are metrics. Define $d(x,y)=d_1(x,y)+d_2(x,y)$.

## Positive-definite

$d(x,y)\geq 0+0=0$.

$d(x,y)=0 \iff d_1(x,y)=d_2(x,y)=0 \iff x=y$.

## Symmetry

$d(x,y)=d_1(y,x)+d_2(y,x)=d(y,x)$.

## Triangle Inequality

\begin{aligned} d(x,y) &\leq d_1(x,z)+d_1(z,y)+d_2(x,z)+d_2(z,y)\\ &=[d_1(x,z)+d_2(x,z)]+[d_1(z,y)+d_2(z,y)]\\ &=d(x,z)+d(z,y) \end{aligned}

Note that it follows by induction that the sum of any finite number of metrics (e.g. three, four or five metrics) is still a metric.