I am fairly new to the concept of supremum/infimum and slightly confused over idea of uniform convergence using these notations. According to the wiki for uniform convergence we say that any $f_n(x)$ uniformly converges to $f(x)$ if $\lim\limits_{n\to\infty} sup|(f_n(x)-f(x))| = 0$.
Now my question is that for $f_n(x) = x^n(1-x)^n$ over [0,1] , If i take $f(x)$ = 1/4 how can I show that this equation holds. What I can see though that $\lim\limits_{n\to\infty} sup(|f_n(x)|) - f(x) = 0$ but won't value of $\lim\limits_{n\to\infty} sup|(f_n(x)-f(x))|$ will be equal to 1/4?