updates

content/Math4111/Math4111_L4.md

@@ -27,7 +27,8 @@
 
 (Archimedean property) If $x,y\in \mathbb{R}$ and $x>0$, then $\exists n\in \mathbb{N}$ such that $nx>y$.
 
-Proof
+<details>
+<summary>Proof</summary>
 
 Suppose the property is false, then $\exist x,y\in \mathbb{R}$ with $x>0$ such that $\forall v\in \mathbb{N}$, nx\leq y$
 
@@ -39,7 +40,7 @@ This implies $(m+1)x>\alpha$
 
 Since $(m+1)x\in \alpha$, this contradicts the fact that $\alpha$ is an upper bound of $A$.
 
-QED
+</details>
 
 ### $\mathbb{Q}$ is dense in $\mathbb{R}$
 
@@ -51,7 +52,8 @@ $$
 x<\frac{m}{n}<y\iff nx<m<ny
 $$
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Let $x,y\in\mathbb{R}$, with $x<y$. We'll find $n\in \mathbb{N},\mathbb{m}\in \mathbb{Z}$ such that $nx<m<ny$.
 
@@ -59,7 +61,7 @@ By Archimedean property, $\exist n\in \mathbb{N}$ such that $n(y-x)>1$, and $\ex
 
 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 $ny>1+nx\geq 1+(m-1)=m$
 
-QED
+</details>
 
 ### $\sqrt{2}\in \mathbb{R}$, $(\sqrt[n]{x}\in\mathbb{R})$