Sorry for this very simple question, but it gets me really confused.
So far, I have perceived the sequences in $\Bbb{R}^n$ as sequences where every element is a point with $n$ numbers. However, I have recently started to get confused, so I felt like I need to be confirmed to be sure that I understand the concepts and the notations correctly.
$(x_1,x_2,x_3,...)$ is an infinite sequence in $\Bbb{R}$ if every $x_i$ is in $\Bbb{R}$- that is $\Bbb{R}^1$. However, I suppose, it becomes a sequence in $\Bbb{R}^n$ if every $x_i$ is a point like $(y_1,y_2,...,y_n)$ where $y_i$ is in $\Bbb{R}$ for all $i$.
I for some reason got this confusion after I attempted to prove that $\Bbb{R}^n$ endowed with euclidian metric is a complete space- I simply could not figure out what I am supposed to subtract from what when using euclidian metric and how to show that satisfying Cauchy criterion implies convergence.