As a bonus problem, our professor of real analysis asked us to prove that the real numbers (a complete ordered field) cannot be extended into an Archimedean field, with no definition of what he meant by extending.
I have tried using proof by contradiction to show that if we have some set, $\mathbb{R}^{*}$ such that $\mathbb{R}$ is a proper subset of $\mathbb{R}^{*}$, and that $\mathbb{R}^{*}$ forms an Archimedean field, then because for there to be new elements in $\mathbb{R}^{*}$ as opposed to $\mathbb{R}$, they would have to be larger (or smaller) than all the elements of R. But then we would have reached a contradiction with the Archimedean property.
Professor returned this solution and said it's not the right solution (without any further comment). Can anyone offer some enlightenment on what I have done wrong or what I could try now?
All I have found on Google are mentions in textbooks that go along the lines of "every Archimedean field is isomorphic to a subfield of real numbers". That would imply that we cannot extend reals into an Archimedean field, but how can one go about proving that?
I can only use basic definition of an ordered (Archimedean) field and other "basics", we have not yet covered sequences, etc.