Please use "proof by contradiction" to prove each of the following claims. Each claim must be proved independently. If both sets contain n elements, The Gale-Shapley algorithm always results in n...

Extracted text: Please use "proof by contradiction" to prove each of the following claims. Each claim must be proved independently. If both sets contain n elements, The Gale-Shapley algorithm always results in n pairs. The resulting pairs are stable; as in, there are no unstable pairs when the algorithm finishes.