A Uniform Distribution Trick

In order for the distribution to be uniform, we require that the probability $p(n)$ of picking any given positive integer $n$ is constant, i.e., $p(n) = k$ $\forall n \in \mathbb{N}$ for some $k \in [0,1].$

As for any distribution, these probabilities must add to $1.$ But for any $k \gt 0$ the sum of probabilities will be $\displaystyle \lim_{m \rightarrow \infty} mk = \infty,$ and if $k = 0$ then the sum will be $\displaystyle \lim_{m \rightarrow \infty} m \times 0 = 0.$ Thus there exists no such value $k$ to make this a valid distribution, and so we cannot pick a positive integer uniformly at random.

Guy 1: "I've got a neat math trick. First, pick a positive integer!"
(...long time passes...)
Guy 2: "..wait! I'm not done writing it down"