UniformConvergence

A sequence of functions ${f_n}$ converges uniformly to a function $f$ on a set $D$ if, for every positive real number $\epsilon > 0$, there exists a positive integer $N$ such that for all $n > N$ and for all $x$ in $D$, the following inequality holds:

$$|f_n(x) - f(x)| < \epsilon$$

In other words, given any desired degree of accuracy $\epsilon$, you can find an index $N$ beyond which the difference between each function $f_n$ and the limit function $f$ is less than $\epsilon$ for every point in the domain $D$. The key aspect of uniform convergence is that the index $N$ does not depend on the point $x$ in the domain.