No heat engine can be more efficient than a reversible engine operating between the same temperature limits (the temperature of heat addition and the temperature of heat rejection) and all reversible heat engines operating between the same temperature limits have the same efficiency.
Proof of Carnot theorem
t1: Temperature of the source
t2: Temperature of the sink
Q: Heat supplied to the heat engine
Q1: Heat rejected by the reversible heat engine
Q2: Heat rejected by non-reversible heat engine
Wr: Work output of the reversible heat engine
W: Work output of the non-reversible heat engine
t1 > t2
Efficiency of non-reversible heat engine is more than efficiency of reversible heat engine.
In the above diagram we can see there are two heat engines operating between the temperature limits t1 (source) and t2 (sink). They both are taking the same amount of heat input and generating some work.
According to our assumption work output by the non-reversible heat engine (W) is more than the work output of the reversible heat engine (Wr)
W > Wr
Q = Q1 + Wr
Since heat output of the reversible heat engine (now heat pump) and heat input of the non-reversible heat engine are the same, it means we can simply connect then bypassing the source.
Now the source becomes redundant.
We can see that the system is generating work(Wextra) by working on single fixed temperature. Which is a clear violation of second law of thermodynamics.
Wextra = W – Wr
It means that our assumption was wrong and efficiency of a reversible heat engine is always more than a non-reversible heat engine.