Statement:

The cyclic integral of dQ/T for a reversible cycle is zero.

Proof:

Assume an irreversible and a reversible engine. The efficiency of a reversible engine is always greater than that of an irreversible engine.

$ηi ≤ ηr$

$1-dQ_2/dQ_1 ≤ 1- T2/T1$

$dQ_2/T2 ≤ dQ_1 /T1$

$dQ_2/T2 - dQ_1 /T1 ≤ 0$

but Q1 and Q2 are in opposite directions

hence, $dQ_2/T2 + dQ_1 /T1 ≤ 0$

$∫dQ/T ≤ 0$