Compare Isothermal Expansion and Adiabatic Expansion

Let us suppose that for an ideal gas, isothermal and adiabatic expansions have initial volume V_i and pressure be P_i to a common final volume V_f. If P_iso and P_adia are the final pressures, then:

For isothermal expansion
P_i V_i = P_iso V_f

And for adiabatic expansion,
P_i V^γ _i = P_adia (V_f) ^γ

Therefore, V_f/V_i = P_i/P_iso

And V_f/V_i = P_i/P_adia

As for expansion (V_f) > (V_i) and also for all gases γ > 1 , hence

{(V_f/ V_i) ^γ} > {V_f/ V_i}

Therefore {P_i/ P_adia} > {P_i/ P_iso}

Or (P_adia) < (P_iso)