Suppose, $\{X_n\}_{n=1}^\infty$ is a sequence of independently distributed sequence of discrete random variables with parameters $\lambda_n$ for each $n\in\mathbb{N}$. All $X_i$'s follow the same distribution, but with different parameters.
Now, suppose all $X_i$'s take some value $a\in\mathbb{R}$ . Then, I'm trying to show that $$\lim_{n\to \infty} \mathbb{P}(X_n=a) = \mathbb{P}(X_n = a~~\text{eventually} )$$ Is this statement always true ? If yes, can we justify this rigorously ?