uniform convergence

Let $X$ be any set, and let $(Y,d)$ be a metric space.A sequence $f_1,f_2,\mathrm{\dots }$ of functions mapping $X$ to $Y$ is said to beuniformly convergent to another function $f$ if, for each $\epsilon >0$, there exists $N$ such that, for all $x$ and all $n>N$, we have .This is denoted by $f_n\stackrel{𝑢}{\to }f$, or “$f_n\to f$ uniformly” or, less frequently, by $f_n\rightrightarrows f$.