$f_n$ uniform converge to f in $[a,b)$ means it uniform converge in $[a,b]$? I have  the given : $f_n$ uniform converge to $f$ in $[a,b)$  I am asked for the following:


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*if $f_n$ is Continuous for every $n\in\mathbb{N}$ is it true that $f_n$ uniform converge to $f$ in $[a,b]$?

*if both $f$ and $f_n$ are Continuous is it true that $f_n$ uniform converge to $f$ in ?
I not sure I understand how to go from $[a,b)$ to $[a,b]$ thats is the main trick I think required here here 

This is true. In fact, if $F: [a;b] \to \Bbb R$ is any continuous function, then $$\sup_{x \in [a;b]} |F(x)| = \sup_{x \in [a;b)} |F(x)|$$ Now, since uniform convergence is defined using the supremum, there is no distinction between $[a,b]$ and $[a,b)$. In other words
$$\lim_n \sup_{x \in [a;b]} |f_n(x)-f(x)| = \lim_n \sup_{x \in [a;b)} |f_n(x)-f(x)|$$
