Recently, I have learnt about the archimedean property of real number system. part of the proof I found on textbook is the following. Assume archimedean property is wrong, which there doesn't exist any positive integer $n$ such that $nx>y$, then clearly $y$ is the upper bound of the set $A=\{nx: n \in \mathbb{N_+}\}$, so A has the least upper bounded property(why? Is the set $A$ equivalent to $\mathbb{R}$). But I don't understand why can we assume there exists an element in A that is the least upper bound of A, isn't having upper bound not logically equivalent to having least upper bounded property? Can anyone explain to me about it? I know the set can be one-to-one correspondance to $\mathbb{Q}$ in term of cardinality, but does it help?
↧
