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

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

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