update notations and fix typos
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@@ -39,7 +39,7 @@ This implies $(m+1)x>\alpha$
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Since $(m+1)x\in \alpha$, this contradicts the fact that $\alpha$ is an upper bound of $A$.
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EOP
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QED
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### $\mathbb{Q}$ is dense in $\mathbb{R}$
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@@ -59,7 +59,7 @@ By Archimedean property, $\exist n\in \mathbb{N}$ such that $n(y-x)>1$, and $\ex
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So $-m_2<nx<m_1$. Thus $\exist m\in \mathbb{Z}$ such that $m-1\leq nx<m$ (Here we use a property of $\mathbb{Z}$) We have $ny>1+nx\geq 1+(m-1)=m$
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EOP
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QED
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### $\sqrt{2}\in \mathbb{R}$, $(\sqrt[n]{x}\in\mathbb{R})$
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