Proof by contradidction that the mean of a set cannot be greater than the greatest value in that set.

Question

I want to prove that given a set of values $x_1, x_2, ..., x_n$, the mean of those values cannot be greater than the greatest of those values.

Let the mean $\frac{x_1 + x_2 +... + x_n}{n} = a$

Assume that $a > x_1, x_2, ..., x_n$ and let $x_1 + x_2 +... + x_n = b$

Then $b < a \cdot n$

Therefore $\frac{b}{n} < \frac{a \cdot n}{n}$, so that $\frac{b}{n} < a$

But since by definition $\frac{b}{n} = a$, this is a contradction.

QED(?)

Answer 1

This is a "fake" proof by contradiction ,ie you don't need to assume something is false to run the proof.

You are just saying

$x_1 + ... x_n \leq n (\max\{x_1,...,x_n\}) $