Proof of An Absolutely Convergent Series of Real Numbers Converges

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named in the step where it is cited. Throughout, $|\cdot|$ is the absolute value on $\mathbb{R}$, and we write $2$ for $1+1$, so that $2|a_{k}|=|a_{k}|+|a_{k}|$.

Claim 1. Suppose that $\sum_{k=1}^{\infty}a_{k}$ converges absolutely, that is, by Series of Real Numbers §absolute, that $\sum_{k=1}^{\infty}|a_{k}|$ converges.

Step 1 (termwise bounds). Fix $k\in\mathbb{N}$. By claim 3 of Properties of the Absolute Value in an Ordered Field we have $-|a_{k}|\le a_{k}\le|a_{k}|$. Applying claim 3 of Elementary Arithmetic in an Ordered Field to the inequality $-|a_{k}|\le a_{k}$ gives $0\le a_{k}-(-|a_{k}|)=a_{k}+|a_{k}|$. Applying the same claim to the inequality $a_{k}\le|a_{k}|$ gives $0\le|a_{k}|-a_{k}$, and

so a second use of claim 3 of Elementary Arithmetic in an Ordered Field, in the direction from $0\le y-x$ to $x\le y$ with $x=a_{k}+|a_{k}|$ and $y=2|a_{k}|$, gives $a_{k}+|a_{k}|\le2|a_{k}|$. Thus

Step 2 (comparison). The linearity clause Elementary Properties of Series of Real Numbers §linearity takes two sequences whose series converge and a scalar; we apply it with $(|a_{k}|)_{k\in\mathbb{N}}$ in both sequence slots, which is legitimate since $\sum_{k=1}^{\infty}|a_{k}|$ converges by assumption, and with the scalar $\lambda=2$. It gives that the series $\sum_{k=1}^{\infty}2|a_{k}|$ converges. By the comparison test Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, applied to the sequences $(a_{k}+|a_{k}|)_{k\in\mathbb{N}}$ and $(2|a_{k}|)_{k\in\mathbb{N}}$, whose hypotheses are exactly the bounds of Step 1, the series $\sum_{k=1}^{\infty}\bigl(a_{k}+|a_{k}|\bigr)$ converges.

Step 3 (conclusion). By Elementary Properties of Series of Real Numbers §linearity, applied again with $(|a_{k}|)_{k\in\mathbb{N}}$ in both sequence slots and with $\lambda=-1$, the series $\sum_{k=1}^{\infty}\bigl(-|a_{k}|\bigr)$ converges, with sum $-\sum_{k=1}^{\infty}|a_{k}|$. Since

the same clause Elementary Properties of Series of Real Numbers §linearity, applied to the two convergent series produced in Steps 2 and 3 and to any scalar, shows that $\sum_{k=1}^{\infty}a_{k}$ converges.

Claim 2. Suppose again that $\sum_{k=1}^{\infty}|a_{k}|$ converges, and write $L=\sum_{k=1}^{\infty}|a_{k}|$. By Claim 1 the series $\sum_{k=1}^{\infty}a_{k}$ converges, and by Elementary Properties of Series of Real Numbers §linearity, applied with $(|a_{k}|)_{k\in\mathbb{N}}$ in both sequence slots and with $\lambda=-1$, the series $\sum_{k=1}^{\infty}\bigl(-|a_{k}|\bigr)$ converges with sum $-L$. By claim 3 of Properties of the Absolute Value in an Ordered Field we have $-|a_{k}|\le a_{k}$ and $a_{k}\le|a_{k}|$ for every $k\in\mathbb{N}$. Applying the comparison of sums Elementary Properties of Series of Real Numbers §order first to the pair of convergent series $\sum_{k=1}^{\infty}\bigl(-|a_{k}|\bigr)$ and $\sum_{k=1}^{\infty}a_{k}$, and then to the pair $\sum_{k=1}^{\infty}a_{k}$ and $\sum_{k=1}^{\infty}|a_{k}|$, gives

By claim 6 of Properties of the Absolute Value in an Ordered Field this two-sided bound is equivalent to $\bigl|\sum_{k=1}^{\infty}a_{k}\bigr|\le L$, which is the assertion.

Claim 3. Assume that $|a_{k}|\le b_{k}$ for every $k\in\mathbb{N}$ and that $\sum_{k=1}^{\infty}b_{k}$ converges. By claim 1 of Properties of the Absolute Value in an Ordered Field we have $0\le|a_{k}|$, so $0\le|a_{k}|\le b_{k}$ for every $k\in\mathbb{N}$. The comparison test Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, applied to the sequences $(|a_{k}|)_{k\in\mathbb{N}}$ and $(b_{k})_{k\in\mathbb{N}}$, shows that $\sum_{k=1}^{\infty}|a_{k}|$ converges and that

Convergence of $\sum_{k=1}^{\infty}|a_{k}|$ is precisely absolute convergence of $\sum_{k=1}^{\infty}a_{k}$, by Series of Real Numbers §absolute, and the remaining inequality $\bigl|\sum_{k=1}^{\infty}a_{k}\bigr|\le\sum_{k=1}^{\infty}|a_{k}|$ is Claim 2.