Sequences of Real Numbers Built from Given Ones: Partial Sums, Index Shifts, Predecessor Reindexing, Absolute Values and Termwise Operations

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $(a_{k})$ and $(b_{k})$ be sequences in $\mathbb{R}$, let $p\in\mathbb{N}$ and $c\in\mathbb{R}$, and let $q:\mathbb{N}_{0}\to\mathbb{R}$ be a map. Finite sums $\sum_{k=1}^{n}a_{k}$ are as in Sums and Products over a Finite Set and over an Interval (iv) and Sums and Products over a Finite Set and over an Interval (v), $|x|$ is the absolute value of $x\in\mathbb{R}$, and for $k\in\mathbb{N}$, $k-1\in\mathbb{N}_{0}$ is the difference, defined as $1\le k$ by Arithmetic and Order of the Natural Numbers (x).

(i) There is a sequence $(s_{n})$ in $\mathbb{R}$ with $s_{n}=\sum_{k=1}^{n}a_{k}$ for every $n\in\mathbb{N}$; it is the sequence of partial sums of $(a_{k})$, and $s_{1}=a_{1}$ and $s_{n+1}=s_{n}+a_{n+1}$ for every $n\in\mathbb{N}$.

(ii) There is a sequence $(d_{k})$ in $\mathbb{R}$ with $d_{k}=a_{p+k}$ for every $k\in\mathbb{N}$.

(iii) There is a sequence $(r_{k})$ in $\mathbb{R}$ with $r_{k}=q(k-1)$ for every $k\in\mathbb{N}$.

(iv) There is a sequence in $\mathbb{R}$ whose $k$-th term is $|a_{k}|$ for every $k\in\mathbb{N}$.

(v) There are sequences in $\mathbb{R}$ whose $k$-th terms are $a_{k}+b_{k}$, $a_{k}-b_{k}$, $a_{k}b_{k}$, $c\,a_{k}$ and $c$, respectively, for every $k\in\mathbb{N}$.