show $f_n = \frac{1}{n} \chi_{[n,\infty]}$ is monotonically decreasing How do I show $f_n = \frac{1}{n} \chi_{[n,\infty]}$ is monotonically decreasing? I know that $\frac{1}{n}$ is monotonically decreasing, but I am unsure how to show $f_n$ is monotonically decreasing because I don't know how to show it is decreasing with the characteristic function.

It is sufficient to prove $f_{n+1}\leq f_n$ for each $n$.

Let $x \in \Bbb{R}$, then either $x < n$ or $n \leq x < n+1$ or $n+1\leq x.$

If $x
If $n \leq x
if $x \geq n+1$, then $f_{n+1}(x) = \frac{1}{n+1} < \frac{1}{n} = f_n(x).$