Suppose $f(x,y)$ is a continuous function, and $c$ is a given finite constant. Fixing $x$, $f(x,y)$ is strictly increasing with respect to $y$, and there exists a unique $y_x$ such that $f(x,y_x)=c$.
Now consider a series of random variables $\{X_n\}$, and assume that $X_n\to x^*$ a.s., where $x^*$ is a constant. With each $X_n$, we can have a random variable $Y_n$ such that $f(X_n,Y_n)=c$. I can use Lemma 5.10 in the book Asymptotic Statistics to show that $Y_n\to Y$ in probability, where $y^*$ is a constant and satisfies $f(x^*,y^*)=c$.
Since the properties of $f$ look very good to me, I wonder if it is true that $Y_n\to Y$ a.s.?
I deeply appreciate any guidance!