$\displaystyle (u_{n}(x) : n=0,1,...) $ be a sequence of real-valued functions on subset E of R.

suppose for all $\displaystyle x\in E, |u_{n}(x)|\leq M_{n}$

where $\displaystyle \sum_{n=0}^{\infty}{M_{n}} $ converges.

prove that: $\displaystyle \sum_{0}^{\infty}{u_{n}(x)} $ converges uniformly on E.