I have learnt that the probability distribution of a binomial random variable $X \sim$Binomial$(n, \frac \lambda n)$ converges to the Poisson distribution with parameter $\lambda$ as $n$ goes to infinity. My question is, for every Poisson random variable, does there necessarily exist a binomial random variable defined on the same sample space whose distribution converges to it? We know that $X\sim$Binomial$(n, \frac \lambda n)$ converges to $Y \sim$Poisson$(\lambda)$; but when considering $Y \sim$Poisson$(\lambda)$, does there always exist an $X\sim$Binomial$(n, \frac \lambda n)$ defined on its sample space?
On some level, I doubt this claim, as for this to be the case, when we break up the sample points in disjoint events, we must have the same probability of success in each of them; and my course hasn't taught me that such is necessarily the case with Poisson distributions- however on an intuitive level, this seems to be the case, because we could introduce partitions and go backwards to get the binomial. So for every Poisson variable, is it the case that its distribution can be explained as the limit of a binomial on the same sample space?