$(f_n)$ converges uniformly $\iff \|f_n-f\|_\infty=\sup\{|f_n(x)-f(x)|;\ x\in I\}$
Prove that: {$f_n$} converges unifromly to $f$ on $E$ as $n \rightarrow \infty$ if and only if $\lim_{n\rightarrow \infty} ||f_n-f||_u = 0$