I have two statements in mind that taken without further caution could seem contradictory:
So I should be missing a point. Is it that the first statement is only valid for the reals or complexes? (however I have the impression that the proof still holds over the rationals)
Or is it rather than the two notions of equivalence (one with bounds, continuity of the identity; the other with equality up to a certain power) are different? Case in which: why are these two natural, what motivates one in some cases and the other one in others?