Let $\displaystyle x_n = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} - \ln(n)$ where $\displaystyle n = 1,2,3,...$

Prove that the sequence $\displaystyle {x_n}$ converges.

Hint: Show, first, that $\displaystyle x > \ln(1+x)$ for any $\displaystyle x > 0$. Then show that the sequence $\displaystyle {x_n}$ increases and $\displaystyle x_{n+1} - x_n < \frac{1}{2(n+1)^2}$. Use the theorem about the convergence and divergence of p-series to complete the proof.

How should I proceed?