Definition:Uniform Convergence/Real Sequence
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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:
- $\ds \forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N: \sup_{x \mathop \in D} \size {\map {f_n} x - \map f x} < \epsilon$
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
Sources
- 1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): $\S 9.3$: Definition of Uniform Convergence: Definition $9.1$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $8.2$: Definition and examples: Definition $8.2.3$
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.3$: Infinite series of functions: Definition $1.4$