description/proof of that when sum of \(n\) squared non-negative numbers is equal to or smaller than squared non-negative number, sum of numbers is equal to or smaller than \(n\) times number
Topics
About: field
The table of contents of this article
Starting Context
- The reader knows a definition of real numbers field.
Target Context
- The reader will have a description and a proof of the proposition that when the sum of any \(n\) squared non-negative numbers is equal to or smaller than any squared non-negative number, the sum of the numbers is equal to or smaller than \(n\) times the 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:
\(r_1\): \(\in \mathbb{R}\) with \(0 \le r_1\)
...
\(r_n\): \(\in \mathbb{R}\) with \(0 \le r_n\)
\(r\): \(\in \mathbb{R}\) with \(0 \le r\)
//
Statements:
\(\sum_{j \in \{1, ..., n\}} r_j^2 \le r^2\)
\(\implies\)
\(\sum_{j \in \{1, ..., n\}} r_j \le n r\)
//
2: Proof
Whole Strategy: Step 1: see that for each \(j \in \{1, ..., n\}\), \(r_j \le r\); Step 2: do \(\sum_{j \in \{1, ..., n\}} r_j \le r + ... + r\).
Step 1:
From \(\sum_{j \in \{1, ..., n\}} r_j^2 \le r^2\), for each \(j \in \{1, ..., n\}\), \(r_j^2 \le r^2\).
As \(0 \le r_j, r\), \(r_j \le r\).
Step 2:
Let us do \(\sum_{j \in \{1, ..., n\}} r_j \le r + ... + r = n r\).
3: Note
As an example, \(r_1 = 1 / 2, r_2 = 1 / 2, r = 1 / 2\): \(r_1^2 + r_2^2 = (1 / 2)^2 + (1 / 2)^2 = 1 / 4 + 1 / 4 = 1 / 2 \le 1 / 2 = r\); \(r_1 + r_2 = 1 / 2 + 1 / 2 = 1 \le 1 = 2 \text{ } 1 / 2 = n r\).
