In the book I am reading, there is this question that says the following: --------- What is wrong with the following proof that 1 is the largest integer? Proof: Let n be the largest integer. Then, since 1 is an integer we must have [itex]1 \leq n[/tex]. On the other hand, since [itex]n^2[/itex] is also an integer we must have [itex]n^2 \leq n[/itex] from which it follows that [itex]n \leq 1[/tex]. Thus, since [itex]1 \leq n[/tex] and [itex]n \geq 1[/tex] we must have n = 1. Thus 1 is the largest integer as claimed. ---------- I know the proof should not be valid, as it is obviously not true, and in fact in the previous question was I had to prove that there does not exist a largest integer, I just cannot find what is wrong in this proof. Now, the last two sentences are valid in my opinion, if we assume that the previous part of the proof is true, so I am thinking it has nothing to do with those last two sentences. The part I don't like is where [itex]n^2[/itex] is introduced and the dividing by n that follows. This is where I feel the proof becomes invalid, but I cannot find a reason for this yet. Any ideas? Thanks!