What would be an easy example of a sequence of functions defined on a compact interval so that $f_n$ goes to $f$ pointwise but $\sup f_n$ does not go to $sup f$.
I thought of the usual example we take to show that the limits in integration can't be interchanged when we only have pointwise convergence. Is this correct?
Does $f(x)=x^n$ work in this context? Any comments or hints?