Is this how I should complete my proof:
By way of contradiction assume that 2k is the largest even integer.
Now consider (2k)!
(2k)! = (2k)(2k-1)(2k-2).......(k)(k-1)(k-2)......(2)(1)
= 2[k(2k-1)(2k-2).......(k)(k-1)(k-2)......(2)(1)]
= 2p, which is also an even integer. This contradicts...