Originally Posted by

**frankmelody** Try to help me to determine whether the following statement is true or not (if true, prove; if false, give a counterexample):

Let $\displaystyle f_j$ be a sequence of functions on a domain $\displaystyle U$, and suppose that $\displaystyle \limsup_{j\rightarrow \infty}{f_j(x)}\leq C<\infty$ for all $\displaystyle x\in U$, then for each $\displaystyle \epsilon>0$, there exists a $\displaystyle J$, such that $\displaystyle f_j(x)\leq C+\epsilon$ for all $\displaystyle j\geq J$ and all $\displaystyle x\in U$.

Thanks!