If a < b are two positive real numbers, let a₀ = a, b₀ = b and $a_{n+1}=\frac{a_n+b_n}{2},b_{n+1}=\sqrt{a_n{b}_n}$ for n≥0. Then (a_n) is increasing and (b_n) is decreasing and $\underset{n\to \infty }{\text{lim}}\left(a_n\right)=\underset{n\to \infty }{\text{lim}}\left(b_n\right)$ is the arithmetic-geometric mean.