Let $x_1, x_2, . . . , x_n$ be $n$ real numbers. Let the average of $$x = \dfrac{x_1 + x_2 + \cdots + x_n}{n}$$ be their average. Prove that at least one of $x_1, x_2, \cdots , x_n$ is greater than or equal to $x$.
I am pretty sure this proof can be proved with contrapositive and I think I may know how to do that. However, I am wondering how you can do it with a contradiction. Any assistance would be greatly appreciated as I am new to discrete math. Thank you for all the comments in advance!
Assume that $x_i<x$ for every $i$. Then: $$x=\frac{x_1+...+x_n}{n}<\frac{x+...+x}{n}=\frac{nx}{n}=x $$ Which is a contradiction