$\displaystyle f_n(x)=1,1\leq x\leq n\\$

$\displaystyle f_n(x)=0,1< n< \infty$

f_n converges to f which is 1

at the beggining f_n is 0 but when n goes to infinity its 1

so why sup(f_n(x)-f(x))=1 ?

f is allways 1

but f_n is 0 and going to one

so the supremumum of their difference is 0 not 1

?