Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 201 / 3 / b / Solution

Put $z=y/\lambda$. Direct integration givesFor $|z|\ge 1$, the right side has a positive infimum $c_0$ because $\sin z/z<1$ there and it tends to one at infinity. Thus the claimed inequality holds with $C=c_0^{-1}$.