Definition:Uniform Convergence

Definition
Let $S$ be a set.

Let $\left({M, d}\right)$ be a metric space.

Let $\left \langle {f_n} \right \rangle$ be a sequence of functions $f_n : S \to M$.

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

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

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

Also see

 * Convergent Sequence

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.