$b_n$ is decreasing and $\lim_{n \to \infty} b_n = 0$

$b_n$ is increasing and $\lim_{n \to \infty} b_n = 0$

$b_n$ is decreasing and $\sum b_n$ converges

$b_n$ is bounded and $\lim_{n \to \infty} b_n = 1$