Order and Arithmetic
Why
How does arithmetic preserve order?
Results
The following are standard useful results.1
If $m < n$, then $m + k < n + k$ for
all $k$.
If $m < n$ and $k \neq 0$, then $m \cdot
k < n \cdot k$.
If $E$ is a nonempty set of natural numbers,
there exists $k \in E$ such that $k \leq m$
for all $m \in E$.
If $E$ is a nonempty set of natural numbers,
there exists $k \in E$ such that $m \leq k$
for all $m \in E$.
- The accounts of which will appear in future editions. ↩︎