Without Loss of Generality

In our proof of the Interior Extremum Theorem, we advanced our argument by supposing that in order to find a sign for so that the order limit theorem could be used. In doing this we put a restriction on the element in our proof that is not necessarily true for all $lates x\in (a,b)$. To complete the proof, we therefore need to show that we have not lost generality by doing this, and our conclusion holds even when .

One way to do this would be to repeat the proof, replacing  with  in the second part, but then the proof would not be concise.

A better way to do this would be to state that  without loss of generality and provide a short justification of why generality is not lost before continuing the proof under the assumption . Another way that we could have written the “without loss of generality” statement in the proof of theorem 5.1.5 is “It is sufficient to prove that for , since when , , so the is still negative. Thus, it follows that  is also true when .”

Another way the phrase “without loss of generality” is used in proofs is to point out when new observations actually add no new restrictions to the proof.