$f_n=\frac{2+(-1)^n}{n}1_{[0,n]} (x) $ and usage of Lemma of Fatou

Question

Given the sequence: $f_n=\frac{2+(-1)^n}{n}1_{[0,n]} (x) $

I have to show whether the following inequations are correct:

1. $\displaystyle\lim\inf_{n\to\infty} \int f_n d\lambda \ge \int \lim\inf_{n\to\infty} f_n d\lambda$

2. $\displaystyle\lim\sup_{n\to\infty} \int f_n d\lambda \le \int \lim\sup_{n\to\infty} f_n d\lambda$

the non-negativity condition of the lemma is met.

So where do we have problems here? I would say both inequalities hold

Answer 1

2. is false. RHS is $0$ because $f_n(x) \to 0$ for every $x$. But $\int f_n(x)dx=2+(-1)^{n}$ so LHS is $3$. .