Show That The Average Translational Kinetic Energy Of The Molecules Of A Gas Is Directly Proportional To Absolute Temperature

Starting from the expression

show that the Average translational kinetic energy of the molecules of a gas is directly proportional to absolute temperature.
Proof
According to the kinetic equation of pressure of a gas:

But r = density of gas
r = density of gas = mass of gas / volume of gas
r = density of gas = mN / V
Putting the value of density (r)

But PV = nRT
putting the value of PV, we get,

Since
number of mole (n) = molecules/Avogadro’s number
number of mole (n) = N/NA
Therefore,

But NA R = Boltzman’s constant (K), thus

Multiplying both sides by 1/2