Equicontinuous Families of Functions
Definition 7.19 Let $\{f_n\}$ be a sequence of functions defined on a set $E$. We say that $\{f_n\}$ is pointwise bounded on $E$ if the sequence $\{f_n(x)\}$...
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Uniform Convergence, Integration, and Differentiation
Theorem 8.4.1 Suppose $f_n\in \mathscr{R}[a,b]$ for all $n\in\mathbb{N}$ ans suppose that the sequence $\{f_n\}$ converges uniformly to $f$ on $[a,b]$. Then ...
Uniform Convergence and Continuity
Theorem 8.3.1 Suppose $\{f_n\}$ is a sequence of real-valued functions that converges uniformly to a function $f$ on a subset $E$ of $\mathbb{R}$. Let $p$ be...
Pointwise Convergence and Uniform Convergence
Definition 8.1.1 (Pointwise Convergence) A sequence of real valued functions $\{f_n\}$ defined on a set $E (\subset \mathbb{R})$ converges pointwise on $E$ i...
Dirichlet Test and Absolute Convergence
Theorem 7.2.1 Let $\{a_k\}$ and $\{b_k\}$ be a sequence of real numbers. Set $A_0:=0, A_n:=\sum_{k=1}^n a_k$ for $n\geq 1$. Then if $1\leq p \leq q$,