Definition:Uniform Convergence/Real Sequence

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

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

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

Also defined as
Some sources insist that $N \in \N$ but this is unnecessary and makes proofs more cumbersome.

Also see

 * Convergent Sequence