change epsilon expression

pages/Math4111/Math4111_L13.md

@@ -106,19 +106,19 @@ To avoid confusion with sets, we use $(p_n)_{n=1}^\infty$ or $(p_n)$
 
 Let $(X,d)$ be a metric space. Let $(p_n)$ be a sequence in $X$.
 
-Let $p\in X$. We say $(p_x)$ **converges** to $p$ if $\forall \varepsilon>0,\exists N\in\mathbb{N}$ such that $\forall n\geq N$, $d(p_n,p)<\varepsilon$. ($p_n\in B_\varepsilon (p)$)
+Let $p\in X$. We say $(p_x)$ **converges** to $p$ if $\forall \epsilon>0,\exists N\in\mathbb{N}$ such that $\forall n\geq N$, $d(p_n,p)<\epsilon$. ($p_n\in B_\epsilon (p)$)
 
 Notation $\lim_{n\to \infty} p_n=p$, $p_n\to p$
 
 We say $(p_n)$ converges if $\exists p\in X$ such that $p_n\to p$.
 
-i.e. $\exists p\in X$ such that $\forall\varepsilon>0,\exists N\in\mathbb{N}$ such that $\forall n\geq N,d(p_n,p)<\varepsilon$
+i.e. $\exists p\in X$ such that $\forall\epsilon>0,\exists N\in\mathbb{N}$ such that $\forall n\geq N,d(p_n,p)<\epsilon$
 
 We say $(p_n)$ **diverges** if $(p_n)$ doesn't converge.
 
 i.e. $\forall p\in X$, $p_n\cancel{\to} p$
 
-i.e. $\forall p\in X$ such that $\exists \varepsilon>0,\forall N\in\mathbb{N}$ such that $\exists n\geq N,d(p_n,p)\geq\varepsilon$
+i.e. $\forall p\in X$ such that $\exists \epsilon>0,\forall N\in\mathbb{N}$ such that $\exists n\geq N,d(p_n,p)\geq\epsilon$
 
 #### Definition 3.2
 
@@ -128,16 +128,16 @@ Example:
 
 $X=\mathbb{C}$, $s_n=\frac{1}{n}$
 
-Then $s_n\to 0$ i.e. $\forall \varepsilon>0 \exists N\in \mathbb{N}$ such that $\forall n\geq N$, $|s_n-0|<\varepsilon$.
+Then $s_n\to 0$ i.e. $\forall \epsilon>0 \exists N\in \mathbb{N}$ such that $\forall n\geq N$, $|s_n-0|<\epsilon$.
 
 Proof:
 
-Let $\varepsilon >0$ (arbitrary)
+Let $\epsilon >0$ (arbitrary)
 
-Let $N\in \mathbb{N}$ be greater than $\frac{1}{\varepsilon}$ (by Archimedean property) e.g. $N=\frac{1}{\varepsilon}+1$ (we choose $N$)
+Let $N\in \mathbb{N}$ be greater than $\frac{1}{\epsilon}$ (by Archimedean property) e.g. $N=\frac{1}{\epsilon}+1$ (we choose $N$)
 
 Let $n\geq N$ (arbitrary)
 
-Then $|s_n-q|=\frac{1}{n}\leq \frac{1}{N}\leq \varepsilon$
+Then $|s_n-q|=\frac{1}{n}\leq \frac{1}{N}\leq \epsilon$
 
 EOP