I have a little bit of confusion. It's hard for me to figure out the difference between using the $\epsilon-\delta$ definition in $\mathbb{R}$ and the $\epsilon-r$ definition in $\mathbb{R}^n$
For example, if I had the problem, show: $\lim_{x\to 0}x^2 = 4$
Using the regular $\epsilon-\delta$ definition, I would do this:
$\qquad$ Let $\epsilon>0.$
$\qquad$ $\lvert x-2\rvert<\delta$. Stipulate $\delta<1$. Now, $\lvert x-2\rvert<1 \implies 1<x<3$. Hence:$\ 3<x+2<5$
$\qquad$ Take $\lvert x^2 -4 \rvert = \lvert (x-2)(x+2) \rvert$
$\qquad$ Now, clearly: $\lvert (x-2)(x+2) \rvert < 5\lvert x-2 \rvert$. Note, we want: $\lvert x-2\rvert<\epsilon/5$. So, $\delta = \epsilon/5$
$\qquad$ Set $\delta = min\{1,\epsilon/5\}$. Then, $\lvert x^2-4\rvert = \lvert (x-2)(x+2) \rvert < 5\lvert x-2 \rvert < (5)(\epsilon/5) = \epsilon$
My question is, if I were using the $\epsilon-r$ definition, would it be okay to just replace $\lvert * \rvert$ with $\left\lVert *\right\rVert$ and $\delta$ with $r$ and call it a day? Thanks ahead of time for the info!