Proof of Elementary Properties of Series of Real Numbers: Linearity, Null Terms, the Cauchy Criterion, Index Shifts, Tails, Order and Telescoping

Each result cited is universally quantified over the data in its own statement. Two auxiliary facts come first.

(P1) Splitting. Let $(d_{k})$ be a sequence in $\mathbb{R}$ and $m,q\in\mathbb{N}$. By Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting §segment, $[m+q]\subseteq\mathbb{N}$; so is the interval $\{m+1,\dots,m+q\}$ of Intervals of Natural Numbers §interval, whose elements $k$ satisfy $1\le m+1\le k$. By Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms §splitting, applied with $m$, $q$ for its $n$, $m$ to the restriction of $(d_{k})$ to $[m+q]$ under the associative addition of $\mathbb{R}$, and read as sums over intervals by Sums and Products over a Finite Set and over an Interval §intervals, $\sum_{k=1}^{m+q}d_{k}=\sum_{k=1}^{m}d_{k}+\sum_{k=1}^{q}d_{m+k}$; and by Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals §interval-shift, applied with $1$, $q$, $m$ for its $m$, $n$, $p$, $\sum_{k=1}^{q}d_{m+k}=\sum_{k=m+1}^{m+q}d_{k}$. Hence

$$\sum_{k=1}^{m+q}d_{k}=\sum_{k=1}^{m}d_{k}+\sum_{k=m+1}^{m+q}d_{k},\qquad\sum_{k=m+1}^{m+q}d_{k}=\sum_{k=1}^{q}d_{m+k}.$$

(P2) Shifting back. Let $(x_{n})$ be a sequence in $\mathbb{R}$, $q\in\mathbb{N}$ and $L\in\mathbb{R}$ with $(x_{q+n})_{n\in\mathbb{N}}$ converging to $L$; then $x_{n}\to L$. Indeed, let a real $\varepsilon>0$ be given; for this $\varepsilon$ choose, by Convergent Sequences of Real Numbers §converges, $N\in\mathbb{N}$ with $|x_{q+n}-L|<\varepsilon$ for every $n\ge N$. Let $n\in\mathbb{N}$ with $n\ge q+N$. As $q<q+N\le n$, $n=q+d$ for some $d\in\mathbb{N}$ by Arithmetic and Order of the Natural Numbers §difference, and $q+N\le q+d$ gives $N\le d$ by Arithmetic and Order of the Natural Numbers §order. Hence $|x_{n}-L|=|x_{q+d}-L|<\varepsilon$. Since $\varepsilon$ was arbitrary and $q+N\in\mathbb{N}$, $x_{n}\to L$ by Convergent Sequences of Real Numbers §converges.

Clause linear. Let $A=\sum_{k=1}^{\infty}a_{k}$ and $B=\sum_{k=1}^{\infty}b_{k}$; then $s_{n}\to A$ and $t_{n}\to B$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges and The Limit of a Convergent Sequence §limit. For $n\in\mathbb{N}$, by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §partial-sums the $n$-th partial sum of $(a_{k}+b_{k})$ is $\sum_{k=1}^{n}(a_{k}+b_{k})=s_{n}+t_{n}$, by Sums and Products over a Finite Set and over an Interval §intervals and Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals §termwise, and that of $(\lambda a_{k})$ is $\sum_{k=1}^{n}\lambda a_{k}=\lambda s_{n}$, by Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §distributive. By Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic, $s_{n}+t_{n}\to A+B$ and $\lambda s_{n}\to\lambda A$. Hence both series converge by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges, with sums $A+B$ and $\lambda A$ by The Limit of a Convergent Sequence §limit.

Clause terms-null. Let $S=\sum_{k=1}^{\infty}a_{k}$, so $s_{n}\to S$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges and The Limit of a Convergent Sequence §limit, and $(s_{n+1})_{n\in\mathbb{N}}$ converges to $S$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §tails. For $n\in\mathbb{N}$, $s_{n+1}=s_{n}+a_{n+1}$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §partial-sums and Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals §interval-recursion, so $a_{1+n}=a_{n+1}=s_{n+1}-s_{n}$. By Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic, $s_{n+1}-s_{n}\to S-S=0$. Thus $(a_{1+n})_{n\in\mathbb{N}}$ converges to $0$, and by (P2) with $q=1$, $(a_{k})$ is a null sequence in the sense of Convergent Sequences of Real Numbers §converges.

Clause cauchy. For $m,n\in\mathbb{N}$ with $m<n$ there is $q\in\mathbb{N}$ with $n=m+q$ by Arithmetic and Order of the Natural Numbers §difference, so (P1) with $(a_{k})$ for $(d_{k})$ and Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §partial-sums give $s_{n}=s_{m}+\sum_{k=m+1}^{n}a_{k}$, that is, $(\ast)$ $\sum_{k=m+1}^{n}a_{k}=s_{n}-s_{m}$. By Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges and Completeness of the Real Numbers for Sequences: Monotone Convergence, the Bolzano-Weierstrass Theorem and Cauchy Sequences §cauchy, the series converges if and only if $(s_{n})$ is a Cauchy sequence as in Cauchy Sequences of Real Numbers §cauchy. Suppose it is, and let a real $\varepsilon>0$ be given; for this $\varepsilon$ choose $N$ by Cauchy Sequences of Real Numbers §cauchy. For $m,n\in\mathbb{N}$ with $N\le m<n$, also $N\le n$, so $\big|\sum_{k=m+1}^{n}a_{k}\big|=|s_{n}-s_{m}|<\varepsilon$ by $(\ast)$. Conversely, suppose the condition of the clause, and let a real $\varepsilon>0$ be given; for this $\varepsilon$ choose $N$ by the condition. Let $m,n\in\mathbb{N}$ with $m,n\ge N$. By Arithmetic and Order of the Natural Numbers §trichotomy, $m=n$, in which case $|s_{m}-s_{n}|=0<\varepsilon$; or $m<n$, in which case $|s_{m}-s_{n}|=|s_{n}-s_{m}|=\big|\sum_{k=m+1}^{n}a_{k}\big|<\varepsilon$ by $(\ast)$; or $n<m$, in which case $|s_{m}-s_{n}|=\big|\sum_{k=n+1}^{m}a_{k}\big|<\varepsilon$ by $(\ast)$ with $m$ and $n$ exchanged. So $(s_{n})$ is a Cauchy sequence by Cauchy Sequences of Real Numbers §cauchy. Thus the condition of the clause holds if and only if $(s_{n})$ is a Cauchy sequence, and, by what was shown first, this holds if and only if the series converges; these are the two equivalences of the clause.

Clause shift. For $n\in\mathbb{N}$ let $u_{n}=\sum_{k=1}^{n}a_{p+k}$, the $n$-th partial sum of $(a_{p+k})_{k\in\mathbb{N}}$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §partial-sums. By (P1) with $(a_{k})$, $p$ and $n$ for $(d_{k})$, $m$ and $q$, $s_{p+n}=s_{p}+u_{n}$ for every $n\in\mathbb{N}$. Suppose $\sum_{k=1}^{\infty}a_{k}$ converges, with sum $S$; then $s_{n}\to S$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges and The Limit of a Convergent Sequence §limit, so $(s_{n+p})_{n\in\mathbb{N}}$, which is $(s_{p+n})_{n\in\mathbb{N}}$, converges to $S$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §tails. The constant sequence $(s_{p})_{n\in\mathbb{N}}$ converges to $s_{p}$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §constant, so $u_{n}=s_{p+n}-s_{p}\to S-s_{p}$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic. Hence $\sum_{k=1}^{\infty}a_{p+k}$ converges by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges, with sum $S-s_{p}$ by The Limit of a Convergent Sequence §limit, which is the stated equation. Conversely, suppose $\sum_{k=1}^{\infty}a_{p+k}$ converges, with sum $U$; then $u_{n}\to U$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges and The Limit of a Convergent Sequence §limit, so $s_{p+n}=s_{p}+u_{n}\to s_{p}+U$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §constant and Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic, and $s_{n}\to s_{p}+U$ by (P2) with $q=p$. Hence $\sum_{k=1}^{\infty}a_{k}$ converges by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges, and the equation holds by the first part.

Clause tails. Let $S=\sum_{k=1}^{\infty}a_{k}$, so $s_{q}\to S$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges and The Limit of a Convergent Sequence §limit. For $q\in\mathbb{N}$, clause shift above with $q$ for $p$ shows that $\sum_{k=1}^{\infty}a_{q+k}$ converges, with $\sum_{k=1}^{\infty}a_{q+k}=S-s_{q}$. The constant sequence $(S)_{q\in\mathbb{N}}$ converges to $S$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §constant, so $S-s_{q}\to S-S=0$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic; that is, the sequence of tails is a null sequence by Convergent Sequences of Real Numbers §converges.

Clause order. Let $A=\sum_{k=1}^{\infty}a_{k}$ and $B=\sum_{k=1}^{\infty}b_{k}$; then $s_{n}\to A$ and $t_{n}\to B$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges and The Limit of a Convergent Sequence §limit. For $n\in\mathbb{N}$, $s_{n}\le t_{n}$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §partial-sums and Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §comparison, applied in the ordered field $\mathbb{R}$ to the restrictions of $(a_{k})$ and $(b_{k})$ to $[n]$. Hence $A\le B$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §order with $K=1$.

Clause telescoping. For $n\in\mathbb{N}$, $s_{n}=\sum_{k=1}^{n}(c_{k+1}-c_{k})=c_{n+1}-c_{1}$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §partial-sums and Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §telescoping, applied to the restriction of $(c_{k})$ to $[n+1]$. Suppose $(c_{k})$ is convergent, with limit $C=\lim_{k\to\infty}c_{k}$ as in The Limit of a Convergent Sequence §limit; then $(c_{n+1})_{n\in\mathbb{N}}$ converges to $C$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §tails, so $s_{n}=c_{n+1}-c_{1}\to C-c_{1}$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §constant and Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic. Hence $\sum_{k=1}^{\infty}a_{k}$ converges by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges, with sum $C-c_{1}$ by The Limit of a Convergent Sequence §limit. Conversely, suppose $\sum_{k=1}^{\infty}a_{k}$ converges, with sum $S$; then $s_{n}\to S$ by Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges and The Limit of a Convergent Sequence §limit, so $c_{1+n}=c_{n+1}=s_{n}+c_{1}\to S+c_{1}$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §constant and Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic. By (P2) with $q=1$, $c_{n}\to S+c_{1}$, so $(c_{k})$ is convergent by Convergent Sequences of Real Numbers §converges, and the formula for the sum holds by what was shown first.