Let $(X_t)_{t\ge 0}$ be a Poisson process with unit rate. Hence, $X_{t+h}-X_t$ is Poisson distributed with mean $h$. Then, \begin{align*} \sum_{n=1}^\infty P(|X_{t+1/n}-X_t| > .5) &= \sum_{n=1}^\infty P(X_{t+1/n}-X_t > .5) \\ &= \sum_{n=1}^\infty (1-e^{-1/n}) = \infty \end{align*}
Hence, by Borel Cantelli's lemma $$P(\limsup_{n\to \infty} X_{t+1/n} >X_t +.5) = 1$$ Does this mean that the time instances in which jumps occur, form a dense subset of $\mathbb{R}^+$? I know that this can't be true, but I am not sure how to understand this.