Without loss of generality, what may we validly assume? So I came across the question where it was asked:
Given that $a,b,c $ are positive integers, in a proof to the theorem $a^3 + b^3 + c^3 \geq a^2b + b^2c + c^2a$, we may assume without loss of generality that:

*

*$a \geq c, b \geq c$

*$a \geq b, a \geq c$

*$a \geq b, b \geq c$

*$a \geq c, c \geq b$
I deduced that the correct answer to the question would be (3) since those are the possible cases and if that's not true, we could reorder $a,b,c$ around. Is my idea of it correct?
 A: Echoing the comments above, the given relation is symmetric up to a cycle of $(a,b,c).$

*

*Therefore, without loss of generality, the variable, say, $c$ may be
validly assumed to be the smallest.
[If we have a particular number at position 1, another particular
number at position 2, and a third particular number at position 3, we
can always cycle through $‘abc’\ldots‘bca’\ldots‘cab’$ until $c$'s position matches that of the smallest number.]
So, option $(1)$ is correct.


*Similarly, without loss of generality, the variable, say, $a$ may be
validly assumed to be the biggest. So, option $(2)$ is also correct.


*On the other hand, if our sequence of numbers is $(1,3,7),$ cycling
through them will never result in them being in descending order. This contravenes option $(3),$ which thus is incorrect.
[The structure of the given relation does not give enough freedom for us to validly infer that if it results from the case whereby $a,b,c$ are ordered descendingly, then it automatically results from the remaining cases.]


*Similarly, if our sequence of numbers is $(1,7,3),$ cycling through
them will never sort them as specified in option $(4),$ which thus also is wrong.
