Sequence of Functions

Definition

Suppose $\varnothing \ne \mathcal{D}\subset \mathbb{R}$, and $\forall n\in \mathbb{N},f_n:\mathcal{D}\to \mathbb{R}$, then $(f_n)$ is called a sequence of function on $\mathcal{D}$

Series of Functions

$\sum _{n=1}^{\infty }{f}_n$ where $f_n$ is a term of a sequence of function $(f_n)$

Facts

Uniform Limit Theorem

Definition

Consider metric spaces $(X,d)$ and $(X,d)$, and a sequence of Continuous Function $(f_n)$ between them $f:X\to Y$. If the sequence $(f_n)$ uniformly converges to $f:X\to Y$, then the function $f$ is a Continuous Function.

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