Why does AM-GM give the maximum value of xy when x+y=2?

[Mr Davis 97]

I am little confused when it comes to optimization problems. For example, say we are given that $x+y=2$, and are asked to maximize $xy$. By AM-GM, we have that $xy \le 1$. But why should this indicate that $1$ is the maximum value? Isn't it an equally true statement to claim that $xy \le 2$, since the former interval is contained in the latter?

 

[andrewkirk]

It is equally true, but not as useful. 2 is merely an upper bound, whereas 1 is a least upper bound. In fact it is a maximum, that is achieved when $x=y$. Arithmetic and Geometric Means are identical when all data are the same.

 

[Mr Davis 97]

It is equally true, but not as useful. 2 is merely an upper bound, whereas 1 is a least upper bound. In fact it is a maximum, that is achieved when $x=y$. Arithmetic and Geometric Means are identical when all data are the same.

I think it was the distinction between upper bound and least upper bound that I was looking for.