The expected maximum of n uniforms
For n independent \mathrm{Uniform}(0,1) variables, find \mathbb{E} M_n and show that it tends to the upper endpoint.
Worked answer (click to show)
On [0,1], F_{M_n}(x)=x^n, so the density is n x^{n-1}. Therefore
\mathbb{E} M_n=\int_0^1 x\cdot n x^{n-1}\,dx=\frac{n}{n+1}.
As n\to\infty, this tends to 1, the upper endpoint.