We have: 

\(a^2+b^2+c^2\ge ab+bc+ca\)

The "=" happen when and only when a = b = c

Applying the Cauchy-Schwarz inequality, we have:

\(\dfrac{1}{a^2+2}+\dfrac{1}{b^2+2}+\dfrac{1}{c^2+2}\le\dfrac{\left(1+1+1\right)^2}{a^2+b^2+c^2+2+2+2}\le\dfrac{9}{ab+bc+ca+6}=\dfrac{9}{9}=1\)

So......................