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

Does anyone have a clever and direct proof of

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

• – player3236 Jan 26 at 19:27
• @player3236 Oh-my-god this is so damn ingenious!!! THANK YOU A WHOLE LOT!!! – ago752 Jan 26 at 19:44