Convergence of the integral of $\frac{\cos(x)}{x^2}$

$|a_m-a_n| = \left|\displaystyle \int_{n}^m \dfrac{\cos x}{x^2}dx\right| \leq \displaystyle \int_{n}^m \dfrac{|\cos x|}{x^2}dx \leq \displaystyle \int_{n}^m \dfrac{1}{x^2}dx = \dfrac{1}{n} - \dfrac{1}{m} < \dfrac{1}{n}$. Thus $\{a_n\}$ is a Cauchy sequence, hence converges.