Limit is taken on a Binomial distribution ($p_n,n$) in this exercise but not on the Poisson($\lambda$) where $\lambda = np_n$. What did I miss?

Question

The wording for this exercise is confusing me a bit. Any clarity would be helpful!

Let $X_n$ be Binomial B($p_n,n$) with $\lambda=np_n$ being constant. Let $A_n$ = {$X_n \geq$ 1}, and let Y be Poisson($\lambda$). Show that $\lim \limits_{n \to ∞} P(X_n=j\,|\,A_n) = P(Y = j\,|\,Y \geq 1)$.

How $\lim \limits_{n \to ∞}$ be taken on only one side of that last equation while keeping $\lambda\;(=np_n)$ constant?