Please prove that the lost of KE is the largest when it undergoes perfectly inelastic collision (stick together after collision)?

For perfectly inelastic collision, the coefficient of restitution e = 0. This indicates that the relative velocity between the two colliding objects after impact is zero. In other words, the two objects move with the same velocity Vc (say) after collision.

Loss of kinetic energy E = initial kinetic energy (Ei) - final kinetic energy (Ef)
For a given system, Ei is fixed. Hence, E will achieve the largest value when Ef is minimum. Because kinetic energy is a scalar, the minimum value of which is zero. That is, all initial kinetic energy of the colliding system is completely lost.

But final kinetic energy Ef = (1/2)(m + M)(Vf)^2
where m and M are the masses of the two colliding objects, and Vf is their final common velocity.

When Ef equals zero (i.e. the minimum value of kinetic energy), this implies that (Vf)^2 is zero (because m and M cannot be zero), or Vf = 0.

Therefore, the conclusion is that if kinetic energy loss is largest (i.e. all initial kinetic energy is lost), the final common velocity of the two colliding objects must be zero (i.e. the two objects are at rest after collision). This is a perfectly inelastic collision situation.

Because if any KE can be resumed, the two collide object will get different speed. That means they will separate and not stick together.