# Sequence using Fatou's lemma

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

## 1 Answer

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 $$\lim (2+\frac 1 n)=2$$.

• Thanks for the feedback, I have voted up Aug 18, 2019 at 12:28