Suppose a function $f_n: \mathbb{R}\to [0,1]$. Holds $$ \lim_{n\to \infty}f_n(y_n) = p\,,$$ where $(y_n)$ is a real sequence such that $$ \lim_{n\to \infty} \frac{y_n}{n}=1.$$
Now consider, a sequence of a random variable $Y_n$ such that $$\frac{Y_n}{n}\xrightarrow{\text{a.s.}} 1, \ n\to\infty.$$ The function $f_n(Y_n) = \mathbb{P}(A\mid Y_n), \ n\geq 1$. I want to show that $\mathbb{P}(A)=p$. Here $A$ is any fixed event.
I proceed through the following way. $$ \mathbb{E}(f_n(Y_n)) = \mathbb{P}(A) $$
then we have to show that only, $$ \lim_n \mathbb{E}(f_n(Y_n)) = p $$
To show that I use Fatou's lemma and the property of $f_n$ that is given.
My question what is the use of the condition $\frac{Y_n}{n}\xrightarrow{\text{a.s.}} 1, \ n\to\infty$? Or is there any other way to prove?