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.