Comparison and Absolute Value Bounds for Finite Sums of Real Numbers

Let $\mathbb{R}$ be the real numbers, an ordered field with additive identity $0$ and order $\le$, and write $|t|$ for the absolute value of $t\in\mathbb{R}$. Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $a:[n]\to\mathbb{R}$ and $b:[n]\to\mathbb{R}$ be maps with values written $a_k$ and $b_k$. All sums below are the finite sums of the field $\mathbb{R}$, and index ranges such as $1\le k\le n$ use the order on the natural numbers.

Then the following hold.

1. (Comparison) If $a_k\le b_k$ for every $k\in[n]$, then

2. (Absolute value)