Proving convergence of sequence $a_n = \int^n_1 \frac{\cos x}{x^2} dx$ I have a sequence $(a_n)$ defined by:
$$a_n = \int^n_1 \frac{\cos x}{x^2} dx$$
I want to prove that this sequence converges, and I have been given a hint:

Prove, for $m \geq n \geq 1$ that $|a_m - a_n| \leq n^{-1}$.

After that is proven, then the sequence is Cauchy and must converge.
My problem is that I cannot prove this. I've tried evaluating this integral:
$$ |a_m - a_n| = \left|\int^m_n  \frac{\cos x}{x^2} dx\right|$$
And my hope was that eventually, after application of some known inequalities e.g. $\frac{1}{n} \cos n \leq \frac{1}{n}$ and the triangle law, eventually I'd end up with the result, but I haven't been successful in getting all the terms with $m$ in them to cancel.
Any help would be appreciated.

$\left|\int^m_n  \frac{\cos x}{x^2} dx\right|\le$ $\left|\int^m_n  |\frac{\cos x}{x^2}| dx\right|\lt \left|\int^m_n  \frac{1}{x^2} dx\right|=| \int^m_n d(-\frac{1}{x})|=[-\frac{1}{x}]_n^m=-\frac{1}{m}+\frac{1}{n} $
Hence $(a_n)$ is Cauchy.
