Clarification on a proof of Fatou's Lemma

Question

I am trying this proof of Fatou's lemma.

$$ \underset{k\ge n}\inf f_k \le f_m, \quad \forall m\ge n.$$

We integrate both sides,

$$\int \underset{k\ge n}\inf f_k d\mu\le \int f_m d\mu, \quad \forall m\ge n.$$

Taking the infinimum of both sides with respect to $m$, we obtain

$$\int \underset{k\ge n}\inf f_k d\mu\le \underset{m\ge n}\inf \int f_m d\mu.$$

Continuing we take the limit $\lim_{n \rightarrow \infty}$ and by considering MCT we arrive at the statement of the Theorem.

I understand all points except from the part were we take the infinimum. Why is it not shown in the left hand side of the inequality?

Answer 1

They use the following Lemma

If $A$ is a number and $a_n$ is a sequence such that $A\leq a_n$ for all $n\geq m$. Then $A\leq \inf_{n\geq m} a_n$

In your question $A:=\int \inf_{k\geq n} f d\mu$ (this is independent of $m$) and $a_m:= \int f_m d\mu$.