Answer
The objective is to prove that the product of m and n is even, however the proof starts out by supposing it is even when it should be proved first.
Therefore, the use of "Thus, mn = (2p)(2q + 1) = 2r" in the conclusion is completely unjustified.
Work Step by Step
Let m be an even integer, by definition of even integers, we write m as
→ m=2p
now, let n be an odd integer, by definition of odd integers, we write n as → n=2q+1
consider their product to be,
mn = (2p)(2q+1)
= 4pq + 2p
= 2(pq+p)
as the product of two integers (pq) is odd and by adding an integer (pq) to another integer (p), we obtain another integer (pq+p), so we can say that (pq+p) is also an integer.
let the integer 2(pq+p) = r,
therefore, mn = 2r,
as 2r is an even integer by the definition of even integers, the product of mn is even!
