Prove $\frac{|a-c|}{\sqrt{(1+a^2)(1+c^2)}}\leq\frac{|a-b|}{\sqrt{(1+a^2)(1+b^2)}}+\frac{|b-c|}{\sqrt{(1+b^2)(1+c^2)}}$

Question

Does anyone have a clever and direct proof of

$\frac{|a-c|}{\sqrt{(1+a^2)(1+c^2)}}\leq\frac{|a-b|}{\sqrt{(1+a^2)(1+b^2)}}+\frac{|b-c|}{\sqrt{(1+b^2)(1+c^2)}}$