Let $(x_n)$ and $(y_n)$ be sequences such that $ x_n$ in $[0, 1]$ and $y_n$ in $[1, 2]$

Question

Use the Bolzano-Weierstrass Theorem to show that there are numbers $m_1, m_2, …$ such that $x_{m_j}$ and $y_{m_j}$ converge for j=1,2,…

So I have trouble approaching this problem. I can apply the Bolzano Weierstrass to the sequences separately to get $x_{m_k}$ converging to some $x$ and $y_{m_j}$ converging to some $y$, but this doesn't help.

I have also thought of adding $x_n$ and $y_n$ and subtracting $x_n$ and $y_n$ to get two bounded sequences each of which converges by Bolzano Weierstrass, but not sure this leads anywhere.

How do you approach this problem?

Answer 1

The Bolzano–Weierstrass theorem holds not only for bounded sequences in $\Bbb R$, but also for bounded sequences in the finite-dimensional space $\Bbb R^d$. This can be seen by repeatedly picking convergent subsequences in each coordinate.

In your case ($d=2$) that would work as follows:

First apply Bolzano-Weierstrass to the (bounded) sequence $(x_n)_n$, that gives a convergent subsequence $(x_{n_k})_k$.

Then apply Bolzano-Weierstrass to the (bounded) subsequence $(y_{n_k})_k$ of $(y_n)_n$, that gives a convergent subsequence $(y_{n_{k_j}})_j$. Note that $(x_{n_{k_j}})_j$ is also convergent as a subsequence of $(x_{n_k})_k$.

Therefore, with $m_j = n_{k_j}$, $(x_{m_j})_j$ and $(y_{m_j})_j$ are “common” convergent subsequences of $(x_n)_n$ and $(y_n)_n$ respectively.