# Definition: Pointwise and Uniformly Convergent Sequences of Functions

Let $\mathbb F$ be a either the field of real numbers or the field of complex numbers and let $D\subseteq \mathbb F$ be the domain of infinitely many given functions $f_n:D\to\mathbb F.$ The sequence of functions $(f_n)_{n\in\mathbb N}$ is called:

• pointwise convergent to a function $f:D\to\mathbb F,$ if for every $x\in D$ the real sequence (or the complex sequence) $(f_n(x))_{n\in\mathbb N}$ is convergent1, i.e. for every $x\in D$ and every $\epsilon > 0$ there exists an index $N\in\mathbb N$ such that $$|f_n(x)-f(x)|<\epsilon\quad\forall n\ge N,$$
• uniformely convergent to a function $f:D\to\mathbb F,$ if the sequence $(f_n)_{n\in\mathbb N}$ is pointwise convergent to $f$ and for a given $\epsilon > 0$ the index $N$ does not depend on $x.$ In other words, for every $\epsilon > 0$ there exists an index $N\in\mathbb N$ such that $$|f_n(x)-f(x)|<\epsilon\quad\forall n\ge N,\quad\forall x\in D.$$

Examples: 1
Proofs: 2 3 4 5 6
Propositions: 7 8 9 10 11

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983

#### Footnotes

1. i.e. a convergent real sequence (or (a convergent complex sequence