Let $X_n \sim \text{Poisson}(\lambda_n)$, with $\lambda_n \to \infty$, $u_n$ such that $u_n \to \infty$, and $p_n := P\big( X_n \leq u_n \big)$.
I'm pretty sure that if $\lambda_n \sim u_n$ or $\lambda_n \ll u_n$, then $p_n\to 1$, eventhough I can't prove it. Any hint/proof is welcome.
Also I think that $u_n \ll \lambda_n$ implies $p_n \to 0$, so my next question is : how to prove it ? and how fast does this convergence occur ?