Harmonic Numbers and a Logarithm

Let $f(x) = \ln(1+x) - \dfrac{x(1 + \lambda x)}{1+x}$.

1. If $f(x) \le 0$ for all $x \ge 0$, find the minimum value of $\lambda$. 2. Let $a_n = 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1n$. Prove that