# Approximating by Simple Functions but $F_k(x)$

I am following the proof of approximating point-wise, non-negative measurable functions by simple functions in Stein and Shakarchi (2009).

I am little confused in their initial steps of truncation.

They state:

For $$k\geq1$$, let $$Q_k$$ denote the cube centered at the origin with a side length $$k$$. Then we define, $$F_k(x)=f(x) \space\space\text{if x\in Q_k,f(x)\leq k},\\k\space\space\text{if x\in Q_k,f(x)> k}\\0\space\space \text{otherwise}.$$ $$F_k(x)\rightarrow f(x)$$ as $$k\rightarrow\infty$$ for all $$x$$.

My questions:

(1) Why are we defining $$F_k(x)$$ here. Is this a simple function? Just recalling the definition, I thought the simple function is a finite sum of characteristic functions over a measurable set. I don't see how $$F$$ is related.

(2) Why does $$F_k$$ converge point-wise to $$f$$?

Reference: $$\textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}$$. Elias M. Stein, Rami Shakarchi. Princeton University Press, 2009.

It looks to me like it is a simple function outside of the $$k$$-cube, and it is a bounded, continuous function inside the $$k$$-cube - is that good enough for the proof they are doing?
It does converge pointwise to $$x$$, because for each $$x$$, eventually both of these is true: 1.) $$x$$ is in every $$k$$-cube for every $$k$$ above some $$N_1$$, and $$f(x) < k$$ for every $$k$$ above some $$N_2$$. So, for each $$x$$, eventually $$F_k(x) = f(x)$$.