An Absolutely Convergent Series of Real Numbers Converges

In the setting of The Real Numbers: Standing Notation and Background, let $(a_{k})_{k\in\mathbb{N}}$ and $(b_{k})_{k\in\mathbb{N}}$ be sequences of real numbers, and write $|t|$ for the absolute value of $t\in\mathbb{R}$. Convergence of a series and its sum, and absolute convergence, are as defined there. Then the following hold.

1. (Absolute convergence implies convergence)(i) If $\sum_{k=1}^{\infty}a_{k}$ converges absolutely, then $\sum_{k=1}^{\infty}a_{k}$ converges.

2. (The triangle inequality for series)(ii) If $\sum_{k=1}^{\infty}a_{k}$ converges absolutely, then

$$\Bigl|\sum_{k=1}^{\infty}a_{k}\Bigr|\le\sum_{k=1}^{\infty}|a_{k}| .$$

3. (Domination by a convergent series)(iii) Suppose that $|a_{k}|\le b_{k}$ for every $k\in\mathbb{N}$ and that $\sum_{k=1}^{\infty}b_{k}$ converges. Then $\sum_{k=1}^{\infty}a_{k}$ converges absolutely, and

$$\Bigl|\sum_{k=1}^{\infty}a_{k}\Bigr|\le\sum_{k=1}^{\infty}|a_{k}|\le\sum_{k=1}^{\infty}b_{k} .$$