Uniform convergence

In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence of functions ${\displaystyle (f_{n})}$ converges uniformly to a limiting function ${\displaystyle f}$ on a set ${\displaystyle E}$ if, given any arbitrarily small positive number ${\displaystyle \epsilon }$, an ${\displaystyle N}$ can be found such that each of the functions ${\displaystyle f_{N},f_{N+1},f_{N+2},\ldots }$ differ from ${\displaystyle f}$ by no more than ${\displaystyle \epsilon }$ at every point ${\displaystyle x}$ in ${\displaystyle E}$. Loosely speaking, this means that ${\displaystyle f_{n}}$ converges to ${\displaystyle f}$ at a "uniform" speed on its entire domain, independent of ${\displaystyle x}$.

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