Cantor's Approach

Instead of using dedekind cuts to construct $\mathbb{R}$, you want to use . Any time we prove something about $\mathbb{R}$, we should only need to use $\mathbb{R}$ is an Ordered Field with the least upper bound property. We'll use it to prove the following:

Proof

1. Assume the opposite for contradiction. Then $\mathrm{\forall }n\in \mathbb{N}(n\le x)$. Then $\mathbb{N}$ is bounded above by $x$, so there is a least upper bound for $\mathbb{N}$, namely $sup(\mathbb{N})=x$. Now look at $sup(\mathbb{N})-1$, which is not an upper bound for $\mathbb{N}$ so then $\mathrm{\exists }n\in \mathbb{N}$ such that $sup(\mathbb{N})-1<n$. But then that means that $sup(\mathbb{N})<n+1$ where $n+1\in \mathbb{N}$! This is not a valid upper bound, so then our assumption is false.
2. Comes immediately from (1).
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