This is the inequality I have to prove: $$\left| e^{-x^2/2} - e^{-y^2/2} \right| \leqslant \alpha |x-y|,$$ for $\alpha \in [0,1)$.
I'm struggling on where to go with it, I've tried hyperbolic functions and series expansions, but it's not going anywhere.
Any help would be great!