description/proof of that real number is equal to or smaller than another real number if it is equal to or smaller than latter plus any positive real number
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of real numbers set.
Target Context
- The reader will have a description and a proof of the proposition that any real number is equal to or smaller than any another real number if it is equal to or smaller than the latter number plus any positive real number.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(r_1\): \(\in \mathbb{R}\)
\(r_2\): \(\in \mathbb{R}\)
//
Statements:
\(\forall \epsilon \in \mathbb{R} \text{ such that } 0 \lt \epsilon (r_1 \le r_2 + \epsilon)\)
\(\implies\)
\(r_1 \le r_2\)
//
2: Proof
Whole Strategy: Step 1: suppose that \(r_2 \lt r_1\), and find a contradiction.
Step 1:
Let us suppose that \(r_2 \lt r_1\).
Let us take \(\epsilon = (r_1 - r_2) / 2\), which satisfies \(0 \lt \epsilon\).
Then, \(r_2 + \epsilon \lt r_1\), a contradiction against that \(r_1 \le r_2 + \epsilon\).
So, \(r_1 \le r_2\).
