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For instance, I am trying to prove

"Let ##X## be a totally ordered set and ##\forall U\subseteq X## s.t. ##U\neq\emptyset## has both a max. and min., then ##X## is finite."

The idea of the proof is very intuitive and recursive, but formalizing it is a bit tricky since there's multiple ways of doing it. After writing a quite complicated and incorrect proof. Then someone wrote the following proof

"Assume X is infinite and let ##x_1## be the bottom element of ##X##. Let ##X_1## = ##X## - {##x_1##} and ##x_2## the bottom element of ##X_1##. Continue in this manner creating the set { ##x_j## : j in N }. which has no maximum."

Which made me a bit irritated because I think using the phrase "Continue in this manner" is like "cheating". So how can I take advantage of this style of writing if I am not aware of the boundaries of what I can and cannot do when writing formal proofs in math?