Originally Posted by

**xalk** Right ,uniform convergence of the sequence of the continuous functions {$\displaystyle f_{n}$},over [a,b], to the functrion,g **implies that g is continuous over [a,b]**

**But we are not done yet it remains to be proved that {$\displaystyle f_{n}$} converges to the function ,g w.r.t the supnorm**

Also in the first part of your proof you have proved pointwise convergence and not uniform,since you have proved:

For all xε[a.b] and ε>0,there exists an N ,such that :

for all ,n if $\displaystyle n\geq N$,then $\displaystyle |f_{n}(x)-g(x)|<\epsilon$

and not

Given ε>0 then there exists N ,SUCH that:

For all ,n and** for all,x **if $\displaystyle n\geq N$ and xε[a,b],then $\displaystyle |f_{n}(x)-g(x)|<\epsilon$