Definition:Uniform Convergence

Definition
Let $$\left \langle {f_n} \right \rangle$$ be a sequence of real functions defined on $$D \subseteq \R$$.

Suppose that $$\forall \epsilon > 0: \exists N \in \R: \forall n \ge N, \forall x \in D: \left|{f_n \left({x}\right) - f \left({x}\right)}\right| < \epsilon$$.

Then $$\left \langle {f_n} \right \rangle$$ converges to $$f$$ uniformly on $$D$$ as $$n \to \infty$$.

(See the definition of convergence of a sequence).

Note
Some sources insist that $$N \in \N$$ but this is not strictly necessary and can make proofs more cumbersome.

Comment
Note that this definition of convergence of a function is stronger than that for pointwise convergence, in which it is necessary to specify a value of $$N$$ given $$\epsilon$$ for each individual point.

In uniform convergence, given $$\epsilon$$ you need to specify a value of $$N$$ which holds for all points in the domain of the function.