First of all I don't understand why we need Banach's theorem, as a result I can't make it intuitive for me to understand how it works but I tried to solve an example.
let $f(x) = x^3$
a) for which fixed points will it converge ?
b) choose an interval $I$ such that all conditions for Banach's theorem are fulfilled (for function $f$)
My try :
a) we have 3 roots $( 0, 1, -1)$ with $0$ being a convergent point. (I think I solved a) but I don't understand what kind of interval are they asking about, and how to compute it)
PS: If some of the text doesn't make sense, please let me know, I translated it from another language.