Fréchet limit for a reciprocal integer tail ($\mathbb{P}(M_n/n\le x)⟶e^{-1/x},x>0$)

Let $X_i$ be independent and identically distributed random variables taking positive integer values with $\mathbb{P}(X_i\le k)=k/(k+1)$, and put $M_n={\max }_{i\le n}{X}_i$. For $x>0$, independence givesIndeed $n/(\lfloor nx\rfloor +1)\to 1/x$ and $\log (1-u)/u\to -1$ as $u\downarrow 0$. For $x\le 0$ the probability is zero. The limiting cumulative distribution function is continuous, including at zero, and is a Fréchet distribution with shape one. Its scale grows linearly with sample size even though the individual random variables have infinite expectation.