# Convergence using Riemann integrability [duplicate]

For each positive integer n, let $\gamma_n = 1+\frac12+ \cdots + \frac1n - \int_1^n \frac1x \, dx$. Prove that the sequence $\{\gamma_n\}$ converges.

## marked as duplicate by Omnomnomnom, Trevor Wilson, Daniel Fischer♦, gt6989b, NamasteNov 18 '13 at 19:01

Here's a hint to a possible approach. Consider the sequence $\{a_{n}\}$ with $a_{n}=\left\{\begin{array}{ll} \frac{1}{n} & : n \text{ odd}\\ \int_{n-1}^{n}\frac{1}{x}dx & : n \text{ even}\end{array} \right.$
Can you show that $\{a_{n}\}$ is decreasing?