# Thread: prove uniform convergence to 0 on I

1. ## prove uniform convergence to 0 on I

Suppose that $\displaystyle \sum u_n(x)$ is uniformly convergent on an interval I. Prove that the sequence { $u_n(x)$} is uniformly convergent to zero on I

2. Originally Posted by wopashui
Suppose that $\displaystyle \sum u_n(x)$ is uniformly convergent on an interval I. Prove that the sequence { $u_n(x)$} is uniformly convergent to zero on I
Let $\displaystyle \|\cdot\|_\infty$ denote the infinity norm on $I$, then by assumption for every $\displaystyle \varepsilon>0$ there exists $N\in\mathbb{N}$ such that $n\geqslant m\geqslant N$ implies $\displaystyle \left\|\sum_{k=1}^{n}u_k(x)-\sum_{k=1}^{m}u_k(x)\right\|_\infty<\varepsilon$. Take $n=m+1$ and conclude.