prove that gravitational force is a conservative force

Gravitational force is a conservative force because the work done against gravity is always conserved.

For Example : When we throw an object vertically above to the height of  h metre then the work done against the gravity is

$$\begin{gathered} \mathrm{W}\text{'}=-\mathrm{mgh} \\ \mathrm{where}, \mathrm{mass} \mathrm{of} \mathrm{the} \mathrm{object}=\mathrm{m} \\ \mathrm{height}=\mathrm{h} \\ \mathrm{acceleration} \mathrm{due} \mathrm{to} \mathrm{gravity}=-\mathrm{g} \end{gathered}$$

and when the object falls back from the same height the work done is

$$\begin{gathered} \mathrm{W}"=\mathrm{mgh} \\ \mathrm{Here} \mathrm{the} \mathrm{total} \mathrm{work} \mathrm{done}, \mathrm{W}=\mathrm{W}\text{'}+\mathrm{W} " \\ \mathrm{W}=-\mathrm{mgh}+\mathrm{mgh}=0 \end{gathered}$$

Hence the total work done in travelling certain height and falling back to same place is zero, this is the property of conservative force as the total work done depends only on the initial and final position of the object. Since the distance travelled is zero, the work done must be zero which is the same here.
That 's why gravitational force is a conservative force.