How to show that Carnot engine is more efficient?

[flux!]

Homework Statement

Consider an Ideal gas engine with the following cycle:
i. Isobaric expansion (T1 -> Th)
ii. adiabatic expansion (Th -> T2)
iii. Isobaric compression (T2 -> T_L)
iv. adiabatic compression (T_L -> T1)
a. Find its efficiency
b. When operated in two temperature, show that Carnot engine is more efficient

Homework Equations

Efficiency of the Ideal gas engine: $$e=1-\frac{T_2 - T_L}{T_H - T_1}$$

Efficiency of the Carnot Engine: $$e=1-\frac{T_L}{T_H}$$

The Attempt at a Solution

I try to approximate the efficiency of the Ideal gas when T1 is very close to TH and TL is very close to T2, T1~TH, TL~T2, then subtract it to the Carnot Efficiency, if it turns out to become positive then Carnot Engine is more efficient. However I get an Indefinite for the Ideal gas efficiency since the denominator becomes zero.

 

[TSny]

Useful information: two legs of the cycle for the ideal gas are adiabatic processes between the same pressures.