On the monotonicity of the sequence $\frac{1}{\log^2 n}$ 
How the monotonicity of the following sequence:
  $$
a_n=\frac{1}{\log^2n}
$$can be shown?

Knowing that $\log n$ is increasing, I thought that I could use the inequality $\log(n+1)>\log(n)$ but then I am not sure if I can square the inequality...
Could you also tell me how to show the monotonicity of $a_n$ with induction or with an other way?

You know that $\log(n+1)>\log(n)$. Then $$\log(n+1)\log(n+1)>\log(n)\log(n+1)>\log(n)\log(n),$$
so $\log^2(n+1)>\log^2(n).$ Taking reciprocals and remembering to reverse the inequality, we get 
$$\frac{1}{\log^2(n)}>\frac{1}{\log^2(n+1)}.$$
This shows the sequence is monotonically decreasing. 
