Exercise $2(17$ points$).$ Consider a certain heuristic function h$($n$)$ that satisfies the following properties $($on a state graph where all costs are positive$)$:

$1.$ h$($n$)=0$ if n is a goal state.

$2.$ For any nodes n$,$ m such that an action that takes us from n to m has a cost c$\_($nm$)>0,$ we have that h$($m$)>=$h$($n$)-$c$\_($nm$).$ Prove that h is an admissible heuristic. Hint: use induction on the number of edges of a shortest path.

PLEASE DRAW OUT A STATE GRAPH THAT SATISFIES THE $2$ CONDITIONS ABOVE WHERE ALL COSTS ARE POSITIVE $($DO NOT VERBALLY PROVE, THAT IS NOT WHAT THE QUESTION IS ASKING$)!$