1074: 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|T.B.P.

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

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About: field

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.

r_{1}

\in R

with

0 \leq r_{1}

...

r_{n}

\in R

with

0 \leq r_{n}

r

\in R

with

0 \leq r

//

Statements:

\sum_{j \in {1, . . ., n}} r_{j}^{2} \leq r^{2}

⟹

\sum_{j \in {1, . . ., n}} r_{j} \leq n r

Whole Strategy: Step 1: see that for each

j \in {1, . . ., n}

r_{j} \leq r

; Step 2: do

\sum_{j \in {1, . . ., n}} r_{j} \leq r + . . . + r

.

Step 1:

From

\sum_{j \in {1, . . ., n}} r_{j}^{2} \leq r^{2}

, for each

j \in {1, . . ., n}

r_{j}^{2} \leq r^{2}

.

As

0 \leq r_{j}, r

r_{j} \leq r

.

Step 2:

Let us do

\sum_{j \in {1, . . ., n}} r_{j} \leq r + . . . + r = n r

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 \leq 1 / 2 = r

;

r_{1} + r_{2} = 1 / 2 + 1 / 2 = 1 \leq 1 = 2 1 / 2 = n r

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