converges uniformly ConvergesUniformly 2003-10-15 01:26:30 2006-09-17 09:54:06 Definition Let $X$ be a set, $(Y,\rho)$ a metric space and $\{f_n\}$ a sequence of functions from $X$ to $Y$, and $f\colon X\to Y$ another function. If for every $\varepsilon>0$ there exists an integer $N$ such that \[ \rho(f_n(x),f(x))<\varepsilon \] for all $x\in X$ and all $n>N$, then we say that $f_n$ \emph{converges uniformly} to $f$.