Convergence of a sequence in $\mathbb{R}^p\times\mathbb{R}^q$ and $\mathbb{R}^{p+q}$

Question

Show that a sequence $(x_1^n,x_2^n)$ in $\mathbb{R}^p\times\mathbb{R}^q$ converges to $(x_1,x_2)$ iff the same thing happens when we consider the sequence as belonging to $\mathbb{R}^{p+q}$.

I am not able to understand the notations used, neither how to proceed.
Here are my beginner questions:-

1. Is $(x_1^n)$ a sequence in $\mathbb{R}^p$ ?
2. Why indexing for sequencing is in superscript, instead of subscript (as is usual) ?
3. Is $x_1^1$ a vector in $\mathbb{R}^p$ ?
4. How can $(x_1^n,x_2^n)$ be a sequence in $\mathbb{R}^{p+q}$ ? (since it is a pair).

Also, please provide me pointers to begin the proof.

Answer 1

1) Yes.

2) If you want to consider a sequence in $\mathbb R^{p}$ it is natural to use superscripts because each element of the sequence already has coordinates. We use subscripts for the coordinates.

3)Yes

4) If $x=(x_1,x_2,...,x_p) \in \mathbb R^{p}$ and $y=(y_1,y_2,...,y_q) \in\mathbb R^{q}$ the notation $(x,y)$ is often used as an abbreviation for $(x_1,x_2,...,x_p,y_1,y_2,...,y_q)$ which is an element of $\mathbb R^{p+q}$

For a proof use the following: if $x$ and $y$ are as above then $\|(x,y)\|=\sqrt {\|x\|^{2}+\|y\|^{2}}$.

Hence the distance between $(x_1^{n},y_1^{n})$ and $(x,y)$ is $\sqrt {\|x_1^{n}-x\|^{2}+\|y_1^{n}-y\|^{2}}$ which tends to $0$ iff both $\|x_1^{n}-x\|^{2}$ and $\|y_1^{n}-y\|^{2}$ tend to $0$.

Answer 2

1. Yes, it is.

2. Because they use the subscript to separate the sequence in $\Bbb R^p$ from the sequence in $\Bbb R^q$. They could have used $(x_n, y_n)$, but they went for $(x_1^n, x_2^n)$. I wouldn't have made that choice myself, but ultimately it's about aesthetics, and has no actual mathematical impact.

3. Yes, it is (this feels like a special case of question 1.)

4. For any $n$, $x_1^n$ is a point in $\Bbb R^p$, and as such, it can be characterised by coordinates as a $p$-tuple like $x_1^n = (x_{1, 1}^n, x_{1, 2}^n, \ldots, x_{1, p}^n)$ (sorry about the indexing). Similarily, $x_2^n$ is a $q$-tuple. Put one after the other, and you get a $(p+q)$-tuple of coordinates, which is to say a point in $\Bbb R^{p+q}$. Since this is done for each $n$, you get a sequence.