Is Measureability Necessary for Proof of Pointwise Limit of Simple Functions?

[aaaa202]

http://www.proofwiki.org/wiki/Measurable_Function_Pointwise_Limit_of_Simple_Functions
The following proof is shown in my book too. Basically it states that every measureable function can be written as the pointwise limit of a sequence of simple functions. Now, my problem is I don't see where the measureability of the limit-function is used crucially - so is the measureability really strictly necessary for the proof and if so where is it used? For me the statement might as well be all functions can be written as the pointwise limit of a sequence of simple functions?

 

[Office_Shredder]

By Characterization of Measurable Functions and Sigma-Algebra Closed under Intersection, it follows that:
Aⁿ_n2n={f≥n}
Aⁿ_k={f≥k2⁻ⁿ}∩{f<(k+1)2⁻ⁿ}
are all Σ-measurable sets.
Hence, by definition, all f_n are Σ-simple functions.

If f was not measurable, then you wouldn't know that the f_ns were actually simple functions