How can I show that $E(X_{max})$ is increasing in $n$, where $X_{max} = max \{X_1, \ldots, X_n\}$, and $X_i$'s are $iid$ draws from a distribution with cdf $F(.)$ and pdf $f(.)$ and with support $X_i \in [0, B]$.
Since $X_{max}$ has cdf $F_{max} = F^n$, I tried to show that $$E(X_{max})= \int_0^B{zdF_{max}(z)} = \int_0^B{znF^{n-1}(z)f(z)dz}$$ has a positive first derivative w.r.t $n$: $$\frac{d}{dn}E(X_{max}) = \int_0^B{zF^{n-1}(z)\Big(1+n\ln(F(z))\Big)f(z)dz}.$$ I am stuck here since $ln(F(z))$ can be hugely negative.
Maybe there is another way of tackling the problem which avoids taking derivative.
Define $M_n=\max\{X_1,\dots,X_n\}$. Then $$ M_{n+1}=\max\{M_n,X_{n+1}\}\geq M_n $$ and so $$ \mathbb{E}[M_{n+1}]\geq \mathbb{E}[M_n]. $$