# Give an example of strict inequality in Fatou's Lemma

The fatou's Lemma says

Let $$\left\{ f_n, n = 1,2,...\right\}$$ be a sequence of non-negative measurable functions. Then $$\liminf \int f_n \geq \int \liminf f_n$$

Some hint?

I think this Showing that the Fatou's lemma inequality can be strict. would work, but I don't understand why $$\liminf f_n=0$$ in the example he gives.

$$f_n(x)=\begin{cases}\boldsymbol 1_{[0,1/2]}(x)&n\ \text{odd}\\ \boldsymbol 1_{[1/2,1]}(x)&n\ \text{even}\end{cases}$$