Let ${f_n}\left( x \right):\left[ {a,b} \right] \to \mathbf{R} $ be continuous and assume that this sequence convergence uniformly on the open interval $(a,b)$. Then the sequence convergences pointwise at the endpoints: $x=a$ and $x=b$.
Does this theorem have a name? For some reason I could barely find it mentioned online, even though I find it quite a useful theorem.