Put $z=y/\lambda$. Direct integration gives
$$
\lambda\int_0^{1/\lambda}(1-\cos uy)\,du
=1-\frac{\sin z}{z}.
$$
For $|z|\geq1$, 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 $C=c_0^{-1}$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-201/3/b/solution