How to construct a sequence $\{g_n\}$ of $\mathscr{A}$-measurable simple functions such that $|g_n(x)|=1$ and $\lim_{n\to\infty}g_n(x)f(x)=|f(x)|$?

I am in the middle of proving a result. It would be unnecessary to type the whole thing out. I just want to ask one step where I got stuck on.

So let $(X,\mathscr{A},\mu)$ be a measure space. Suppose that $f$ belongs to $\mathscr{L}^1(X,\mathscr{A},\mu,\mathbb{C})$. How can I construct a sequence $\{g_n\}$ of $\mathscr{A}$-measurable simple functions for which the relations $|g_n(x)|=1$ and $\lim_{n\to\infty}g_n(x)f(x) = |f(x)|$ hold at each $x$ in $X$?

Thank you very much for any help!