Prove Primitive Recursive Functions are Programs with Bounded FOR Loops

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The first sentence of
https://en.wikipedia.org/wiki/Primitive_recursive_function
(Google: primitive recursive function wiki)
is:

In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined before entering the loop).

Prove this and answer my questions and objections about your proof.

Note: I haven’t been to school in decades.
Related categories: Mathematics Computer Science