Let $f(x,y):[0,1]\times[0,\infty)\to[0,1]$ be a continuous function. Let $$F(x)=\limsup_{y\to\infty}f(x,y).$$
Show that $F$ is a measurable function on $[0,1]$.
My attempts
In my limited experience, most of the problems concerning measurability give a sequence and ask about their (upper) limit. This question is rather novel to me. So I decide to return to the definition of a measurable function and consider $F^{-1}((a,b))$.
$$F^{-1}((a,b))=\{x:\limsup_{y\to\infty}f(x,y)\in (a,b)\}$$
I guess the essential of this problem is to apply the countinuity properly when $f$ occurs in a $\limsup$.
THX :)
Verify that $\{x: F(x)<a\}=\bigcup_k \bigcup_n \bigcap_{y\in \mathbb Q, y>n} \{(x: f(x,y) \leq a-\frac 1 k\}$/