Sequence using Fatou's lemma

Question

I have the following problem which I am trying to solve using Fatou's lemma:

Let $(f_n)_{n\geq1}$ be a sequence in $\mathcal{M}^+$ and let $f \in \mathcal{M}^+$. Assume $f_n \rightarrow f$ pointwise. Also assume $\int_X f_n d\mu=2+1/n$. Show $\int_X f d\mu\leq 2$

My approach is: $\int_X f d\mu=\int_X lim_{n\rightarrow \infty} f_n d\mu \leq lim_{n \rightarrow \infty} \int_X f_n d\mu = lim_{n\rightarrow\infty} 2+1/n$

But this has several problems. First of all it doesn't show $\leq2$ but $\leq 2+1/n$. Secondly, I am using lim instead of lim-inf which is the definition for Fatou's lemma in the textbook. Thirdly, I am not sure if I can apply the inquality using Fatou's lemma this way

Answer 1

Just write $\lim \inf $ instead of limit throughout the inequalities. Observe that $\lim \inf f_n =\lim f_n$ because $\lim f_n$ exists by hypothesis. Of course $\lim (2+\frac 1 n)=2$.