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I am trying to understand this remark in a paper by Bollobás and Riordan:

Let $X_1, X_2,\dots$ be the points of a Poisson process on $[0, \infty]$ with rate $m$, so, setting $X_0 = 0,$ the variables $X_i − X_{i−1}$ are iid exponentials with mean $1/m.$ Let $Y_i = \sqrt{X_{mi}},$ and let $D_m = \max\{Y_i − Y_{i−1},1 \leq i < \infty\}$, noting that this maximum exists with probability one.

How do you show the "maximum exists with probability one"?

(Clarification: $m$ is an arbitrary natural number, and $X_{mi}$ means the $(mi)^\text{th}$ point, where $mi=m\text{ times }i$ (in particular, $X_{mi}$ is not "just another name" for $X_i$).)

Bollobás, B., & Riordan, O. M. (2003). Mathematical results on scale-free random graphs. Handbook of graphs and networks: from the genome to the internet, 1-34.

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