Show that if $G$ is a connected graph such that the degree of every vertex is one of 3 distinct number and each of these three number is degree of at least one vertex of $G$, then there is a path in $G$ contain three vertices whose degree are distinct.
I tried to let $a,b,c$ ($a,b,c \geq 1$) be the number of vertices of degree $x,y,z$ respectively for $x,y,z$ are distinct. The book told me that I need to consider the path that contain 2 vertices of distinct degree. But I don't understand how this could help me. Can anyone enlighten me please?