# Thread: Lim sup of a sequence of functions

1. ## Lim sup of a sequence of functions

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!

2. 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!
Suppose the claim were false. Then we would have $\displaystyle f_j(x)>C+\epsilon$ for infinitely many $\displaystyle j$. And so we would be able to find a subsequence such that $\displaystyle f_{j_k}(x)>C$, which would contradict $\displaystyle \limsup_{j}f_j(x)\le{C}$