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.