Sums of Sequences in [0,∞]: Real Series, Linearity, Comparison, Rearrangement, Double Series and Monotone Convergence

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\infty$, $[0,\infty]$ and the order $\le$ be as in The Extended Real Line, Its Order, and [0,∞] (i), The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii), the sum $+$ and the product $\cdot$ as in Addition and Multiplication on [0,∞] (i) and Addition and Multiplication on [0,∞] (ii), $\sup_{n}$ as in Order and Suprema in the Extended Real Line (iii), sums over finite sets of maps into $[0,\infty]$ as in Sums and Products over a Finite Set and over an Interval (i), Sums and Products over a Finite Set and over an Interval (ii), Sums and Products over a Finite Set and over an Interval (iii) and Sums and Products over a Finite Set and over an Interval (v), for $+$, which is associative and commutative with neutral element $0$ by Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (i), and sums of sequences in $[0,\infty]$ as in Sum of a Sequence in [0,∞] (ii). Let $(a_{k})$ and $(b_{k})$ be sequences in $[0,\infty]$, and $s_{n}=\sum_{k=1}^{n}a_{k}$ the partial sums of $(a_{k})$.

(i) $s_{n}\le s_{n+1}$ for every $n\in\mathbb{N}$.

(ii) $s_{n}\le\sum_{j\in\mathbb{N}}a_{j}$ for every $n\in\mathbb{N}$.

(iii) $a_{k}\le\sum_{j\in\mathbb{N}}a_{j}$ for every $k\in\mathbb{N}$.

(iv) For every finite subset $F$ of $\mathbb{N}$, $\sum_{k\in F}a_{k}\le\sum_{k\in\mathbb{N}}a_{k}$.

(v) If $t\in[0,\infty]$ and $\sum_{k\in F}a_{k}\le t$ for every finite subset $F$ of $\mathbb{N}$, then $\sum_{k\in\mathbb{N}}a_{k}\le t$.

(vi) Let $a_{k}\in\mathbb{R}$ for every $k\in\mathbb{N}$. Then $\sum_{k\in\mathbb{N}}a_{k}\in\mathbb{R}$ if and only if the series $\sum_{k=1}^{\infty}a_{k}$ converges, and in that case the two sums are equal; otherwise $\sum_{k\in\mathbb{N}}a_{k}=\infty$. If instead $a_{k}=\infty$ for some $k\in\mathbb{N}$, then $\sum_{k\in\mathbb{N}}a_{k}=\infty$.

(vii) If $n\in\mathbb{N}$ and $a_{k}=0$ for every $k\in\mathbb{N}$ with $n<k$, then $\sum_{k\in\mathbb{N}}a_{k}=s_{n}$.

(viii) $\sum_{k\in\mathbb{N}}(a_{k}+b_{k})=\sum_{k\in\mathbb{N}}a_{k}+\sum_{k\in\mathbb{N}}b_{k}$.

(ix) $\sum_{k\in\mathbb{N}}ca_{k}=c\sum_{k\in\mathbb{N}}a_{k}$ for every $c\in[0,\infty]$.

(x) If $a_{k}\le b_{k}$ for every $k\in\mathbb{N}$, then $\sum_{k\in\mathbb{N}}a_{k}\le\sum_{k\in\mathbb{N}}b_{k}$.

(xi) If $\sigma:\mathbb{N}\to\mathbb{N}$ is a bijection, then $\sum_{k\in\mathbb{N}}a_{\sigma(k)}=\sum_{k\in\mathbb{N}}a_{k}$.

Let $w:\mathbb{N}\times\mathbb{N}\to[0,\infty]$ be a map.

(xii) $\sum_{m\in\mathbb{N}}\sum_{n\in\mathbb{N}}w(m,n)=\sum_{n\in\mathbb{N}}\sum_{m\in\mathbb{N}}w(m,n)$.

(xiii) If $\tau:\mathbb{N}\to\mathbb{N}\times\mathbb{N}$ is a bijection, then

$$\sum_{m\in\mathbb{N}}\sum_{n\in\mathbb{N}}w(m,n)=\sum_{n\in\mathbb{N}}\sum_{m\in\mathbb{N}}w(m,n)=\sum_{k\in\mathbb{N}}w(\tau(k)).$$

(xiv) If $w(n,k)\le w(n+1,k)$ for all $n,k\in\mathbb{N}$, then $\sum_{k\in\mathbb{N}}\sup_{n}w(n,k)=\sup_{n}\sum_{k\in\mathbb{N}}w(n,k)$.