Prove that at least one of the real numbers $a_1 , a_2 , … , a_n$ is greater than or equal to the average of these numbers

Question

Prove that at least one of the real numbers $\,a_1 , a_2 , … , a_n$ is greater than or equal to the average of these numbers. What kind of proof did you use?

I think I should use contradiction but I don't know how should I use that.

Thank you so much.

Answer 1

Let average $g$ and $a_i<g$ for $1\le i\le n$

$$\implies g\cdot n=\sum_{1\le i\le n}a_i<\sum_{1\le i\le n}g=g\cdot n$$

Answer 2

Let $a = \max\limits_{1 \leq i \leq n}a_i$ then you can see that the average $ m = \frac{1}{n}\sum\limits_{i = 1}^n a_i \leq \frac{a}{n}\sum\limits_{i = 1}^n 1 = a, $ because each $a_i$ is less than or equal to a, hence the result.