How obvious must it be before being allowed to use "Without Loss of Generality" in a proof

Question

For example I have a proof where I can only use positive numbers less than or equal to 4 for a variable. The equation just off looking is pretty obvious, if I plugged any number from 1-4 into it then I'd prove the proposition is true. I wanted to give one example, where I assume the variable is 1 show the proposition is true then say WLOG the other 3 cases are true. Is that too much of an over reach in solving proofs?

Answer 1

You are probably ok showing one example and assuming the rest. However where you seem to be going wrong is in saying 'WLOG'. This expression has a specific meaning where symmetry or some similar consideration mean that it is logically valid to skip a case. For example, if there are two interchangeable numbers in your proof, one must be greater than or equal to the other. So we can assume that $a \geq b$, since $b \geq a$ is symmetric.

This only applies when the symmetric cases are interchangeable. It certainly doesn't seem to be the case for 1, 2, 3 or 4. So you should use different language if you are skipping one or more of these.

However, isn't it possible to write a more general proof which works for any number smaller or equal to 4? This is probably a good, readable, approach if possible.