Say $f_n(x)$ (or$ \sum f_n(x)$) : $D \to \mathbb{R}$, Let a bounded interval $I(\subset D)$
First, In the series case
"$\sum f_n(x)$ is a uniformly convergence on $\forall I \Rightarrow$ $\sum f_n(x)$ is a uniformly convergence on $D$"
I've just known that the above is a false statement.
Because I found the counterexample that $f(x) = \sum_{n=1}^{\infty} sin({x \over n^2})$.
It is uniformly converge on $\forall I$ like a $[M_1,M_2]$, But it is not uniformly converge on $\mathbb{R}(=D)$ by taking $x=n^2$
Second, Let me focus on the sequence of the functions.
$(*)$ "$f_n(x)$ is a uniformly convergence on $\forall I \Rightarrow$ $f_n$ is a uniformly convergence on $D$"
My guess is it would be still false when we considering $f_n(x)$ instead of the $\sum f_n(x)$. But unlike the series, I can't find any counterexamples. So my question is Is the statement (*) is true? or not, Please help me for finding any counterexamples.
Any help would be appreciated. thanks.