Definition of Uniformly Convergence of a Sequence of Function

 A sequence of real function (f_n) is said uniformly convergent on a set A⊆ℝ to a limit function f:A→ℝ if and only if for every 𝜺 > 0 there exists N ∈ ℕ such that for every natural number n ≥ N and for every x ∈ A,

|f_n(x) - f(x)| < 𝜺