Fatou's lemma. Prove or disprove.

Question

Let $(f_{n})$ be a sequence of integrable functions on $\mathbb{R}$. Prove or disprove:

1. $$\begin{align} \int_{\mathbb{R}}\lim\inf_{n\rightarrow\infty}f_{n}\leq \lim\inf_{n\rightarrow\infty}\int_{\mathbb{R}}f_{n} \end{align}$$
2. $$\int_{\mathbb{R}}\lim\inf_{n\rightarrow\infty}f_{n}\geq \lim\inf_{n\rightarrow\infty}\int_{\mathbb{R}}f_{n}$$
3. $$\int_{\mathbb{R}}\lim\sup_{n\rightarrow\infty}f_{n}\leq \lim\sup_{n\rightarrow\infty}\int_{\mathbb{R}}f_{n}$$
4. $$\int_{\mathbb{R}}\lim\sup_{n\rightarrow\infty}f_{n}\geq \lim\sup_{n\rightarrow\infty}\int_{\mathbb{R}}f_{n}$$

My works:

1. To show, I provided a counter example $f_{n}=-n1_{[0,1/n]}$. That is , ${\int}_{\mathbb{R}}\underset{n\rightarrow\infty}{\liminf}f_{n} = 0$ and $\underset{n\rightarrow\infty}{\liminf}\int_{\mathbb{R}}f_{n} = -\infty$.

Am I going the correct way?. Thanks in advance.