13.7. Question 6

The question 6 says you have to prove that there exists at least one real number among a set of n real numbers which is greater than equal to their average. I stress here that a to a are arbitrary here. You cannot show concrete values of a to a and prove this statement for those concrete values and conclude that this statement is true.

In this problem, we are tasked with proving that in any collection of n real numbers, at least one of those numbers will be greater than or equal to the average of the entire set. The average is calculated by summing all the numbers and dividing by the count of numbers. Here, 'arbitrary values' means any real numbers can be selected; no specific values are used for this proof. This universality is essential because it underlines the argument's applicability to any potential subset of real numbers.

Imagine you are a teacher with several students in your class, each scoring different marks on a test. The average score represents the typical performance of the whole class. If all students scored below the average, it would mean there's a contradiction, as the averages wouldn't make sense given the scores. Thus, at least one student must have scored at or above that average.

What we do here is we give proof by contradiction. So our goal is to prove that average of a to a is less than equal to some a. But instead, I assume that each of the individual numbers among these n numbers is less than their average.

To prove this statement, we employ a proof by contradiction approach. This means we assume the opposite of what we want to prove—in this case, that every number in our set is less than the average. If this were true and we added all these inequalities together, we would find that the aggregate of all the individual numbers, which constitute the total, would be less than the total itself, thus leading to a logical inconsistency.

Think of a group of friends who all believe they are 'not the best' at a game they play together. If every friend is less skilled than the group's average skill, it paradoxically suggests the average cannot exist, as it implies one friend is better. This contradiction helps us understand that in reality, at least one must meet or exceed that average skill level.

Now if I add this n equations I get this inequality. And if I substitute the value of the average by this formula I come to the conclusion that the summation of n numbers is less than the summation of n numbers which is not possible which is a contradiction.

When we add the individual inequalities together, we'll find that if they all hold true (i.e., that each number is less than the average), the sum total cannot equal what was computed for their average. This is mathematically impossible, as you can’t get a smaller total by adding positive real numbers. Therefore, we conclude that at least one number must equal or exceed the average, because our assumption must be wrong.

Visualize a potluck dinner: if every person brought fewer dishes than the average, the collective would end up with fewer total dishes than the average amount brought. Clearly, this cannot happen unless at least one person brought more than the average amount, thus reaffirming that averages have meaning in a real context.

That means assuming this contradiction leads to a false conclusion that means the statement is a true statement. That means you take any n real numbers, any n arbitrary real numbers, they could be positive, negative, they may be the same, different. At least one of them will be greater than or equal to their average.

Having reached a contradiction by assuming that all numbers in our set were below average, we confirm that at least one of the numbers must indeed be equal to or greater than the average. This conclusion holds true regardless of the nature of the numbers we choose, highlighting a consistent truth about averages.

Consider a company that measures employee productivity across a department. If every employee believes they've performed under average productivity, it suggests a flaw in their metrics, as averages reflect broader success. Therefore, someone in that department must exceed the average to maintain realistic comparisons, affirming that averages accurately reflect group performance.

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