$\liminf_{n_k}$ versus $\liminf_n$ with integrals

I am working with a practice problem which states the following:

Let $(X, \mathcal{F}, \mu)$ be a measure space and let $\{ f_n: X \rightarrow [0, \infty] \}_{n=1}^\infty$ be a sequence of $\mathcal{F}$-measurable functions. Consider the function $f: X \rightarrow [-\infty,\infty]$ which has the property $f_n \xrightarrow \mu f$ ($f_n$ converges to $f$ in measure $\mu$). Show that$$ \int_X f d\mu \leq \liminf_{n \rightarrow \infty} \int_X f_n d\mu .$$

My idea was to use the property that if $f_n$ converges to $f$ in measure, there exists a subsequence such that $f_{n_k} \rightarrow f$$\mu$-almost everywhere. This would then allow me to apply Fatou's lemma so that

$$ \int_X \liminf_{n_k} f_{n_k} d\mu = \int_X f d\mu \leq \liminf_{n_k}\int_X f_{n_k} d\mu$$

This is where I get stuck as I don't know if I can conclude the statement from here or not. Is my work so far correct and if so can anyone help me along the way?