Answer
We have given that: A matching table is stable, when there is no man $m$ and no women $w$ such that $m$ prefers $w$ over his assigned partner and $w$ prefer $m$ over her assigned partner.
$Proof$ $by$ $contradiction$
Let us assume, for the sake of contradiction, that the algorithm does not end with a stable assignment.
Then there exists a man $m$ and a woman $w$ such that $m$ prefers $w$ over his assigned partner and $w$ preferred $m$ over her assigned partner.
This would then mean that $m$ would have proposed $w$ in a previous iteration of the algorithm. Since $w$ prefers $m$ over her assigned partner, she would then have rejected the proposal of the assigned partner in that iteration of the algorithm.
However, this is not possible as she cannot reject a proposal of somebody who became the assigned partner and thus we have derived a contradiction.
This then means that our assumption “the algorithm does not end with a stable assignment” is incorrect and thus the deferred acceptance algorithm terminates with a stable assignment.
Work Step by Step
We have given that: A matching table is stable, when there is no man $m$ and no women $w$ such that $m$ prefers $w$ over his assigned partner and $w$ prefer $m$ over her assigned partner.
$Proof$ $by$ $contradiction$
Let us assume, for the sake of contradiction, that the algorithm does not end with a stable assignment.
Then there exists a man $m$ and a woman $w$ such that $m$ prefers $w$ over his assigned partner and $w$ preferred $m$ over her assigned partner.
This would then mean that $m$ would have proposed $w$ in a previous iteration of the algorithm. Since $w$ prefers $m$ over her assigned partner, she would then have rejected the proposal of the assigned partner in that iteration of the algorithm.
However, this is not possible as she cannot reject a proposal of somebody who became the assigned partner and thus we have derived a contradiction.
This then means that our assumption “the algorithm does not end with a stable assignment” is incorrect and thus the deferred acceptance algorithm terminates with a stable assignment.
