When taking the course of real analysis, the professor wrote the following:
Let $f_n \geq 0$ for all $n$.
Let $$a = \lim \inf \int_X f_n d\mu + \epsilon$$ I think here $\epsilon$ is any positive number. Now, by the definition of $\lim \inf$, we have $$\forall N, \exists\, n >N, \text{ such that } \int_X f_nd\mu <a$$
My question is, if $\{f_n\}$ is decreasing, i.e., $$f_1\geq f_2 \geq f_3 \cdots \geq f_n \geq \cdots,$$ then how could the conclusion hold? (if choosing $\epsilon \rightarrow 0$)