Let $\theta$ be an angle between unit vectors a and $b$.
Then, $ a. b =\cos \theta$
Now, $| a + b |^{2}=| a |^{2}+| b |^{2}+2 a \cdot b$
$=2+2 \cos \theta=4 \cos ^{2} \frac{\theta}{2}$
$| a - b |^{2}=| a |^{2}+| b |^{2}-2 a \cdot b$
$=2-2 \cos \theta=4 \sin ^{2} \frac{\theta}{2}$
$\therefore | a + b |=2 \cos \frac{\theta}{2},| a - b |=2 \sin \frac{\theta}{2}$
$\therefore | a + b |+| a - b |=2\left(\cos \frac{\theta}{2}+\sin \frac{\theta}{2}\right) \leq 2 \sqrt{2}$