\(\dfrac{a^2+b^2}{2}\ge\dfrac{\left(a+b\right)^2}{2^2}\)

<=> \(4\left(a^2+b^2\right)\ge2\left(a+b\right)^2\)

<=> \(4\left(a^2+b^2\right)\ge2\left(a^2+2ab+b^2\right)\)

<=>  \(\left(2a^2+2b^2-4ab\right)\ge0\)

<=> \(2\left(a-b\right)^2\ge0\)   (Right)

So ....