# Thread: Monotone Decreasing

1. ## Monotone Decreasing

Let $(f_n)$ be a non-negative sequence of monotone decreasing integrable functions on $[0,1]$.
Show that $(f_n) \downarrow$ 0 if and only if $(\int_{[0,1]} f_n) \downarrow$ 0.

2. Originally Posted by problem
Let $(f_n)$ be a non-negative sequence of monotone decreasing integrable functions on $[0,1]$.
Show that $(f_n) \downarrow$ 0 if and only if $(\int_{[0,1]} f_n) \downarrow$ 0.
Since the functions are non-increasing (monotome decreasing) it means that $f_n(b) \leq f_n(x) \leq f_n(a)$ for each $x\in [a,b]$. Therefore, $f_n(b) \leq \smallint_0^1 f_n \leq f_n(a)$. However, we are told that $f_n(a),f_n(b)\to 0$ therefore the integral, being squeezed between those two sequences, must go to zero too. Conversely it is not true. Just let $f_n(a) = 1$ with $f_n(x) = 0$ for $a. Then each $f_n(x)$ is integrable, non-negative, non-increasing with zero integral however $f_n$ does not converge to the zero function.

3. the monotone decreasing that i refer to is that $0\le f_{n+1} \le f_n$ for all n.Is it still valid to use the argument above?