Let $f(x) = x - \ln(1 + x)$.
(a) Prove that $f(x) \ge 0$ for all $x > -1$.
(b) By using part (a), or otherwise, determine whether the following integrals are convergent or divergent.
i. $\int_{2}^{\infty} \frac{1}{\ln y} dy$,
ii. $\int_{3}^{\infty} \frac{1}{(\ln w)^{2}} dw$,
iii. $\int_{4}^{\infty} \frac{\ln(z)\sin(z)}{z^{3}} dz$.
