Definition:Uniform Convergence/Real Sequence

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

Let:
 * $\forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N, \forall x \in D: \size {\map {f_n} x - \map f x} < \epsilon$

That is:
 * $\displaystyle \forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N: \sup_{x \mathop \in D} \size {\map {f_n} x - \map f x} < \epsilon$

Then $\sequence {f_n}$ 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

 * Definition:Convergent Sequence