I have the question:
Show that the number of walks of length $2n$ that start at $(0, 0)$ and do not touch the $x$-axis is equal to the number of walks of length $2n$ that hit $0$ at time $2n$.
I have that the number of walks of length $2n$ from $(0,0)$ to $(2n,0)$ is $2n$ Choose $n$. However I don't know where to progress here, I have the positive walks theorem and this need to be multiplied by $2$ to include all walks that are strictly negative but I cant seem to find a proof of this theorem.
Any help is much appreciated