Choose the right proof pattern

Question

I am a bit confused with the following problem.

Let's say, there is a group of 10 people and the task is to prove that at least one of them has the height less or equal to the average height of the group.

I tried to use a proof by contradiction as follows: A height of each person in the group is greater than average (which is obviously false) and therefore it is a contradiction, but not 100% sure if it is the right way to solve this problem.

Could anyone advise me how to prove it?

Answer 1

We have the formula $$ A = \frac {x_1+x_2+...+x_n}{n}$$

Now if for each $n$ we have $x_n>A$, we will get $$A = \frac {x_1+x_2+...+x_n}{n}> \frac {nA}{n}=A$$

That is $A>A$ which we do not accept.

Thus for at least one n, we have to have $x_n\le A$