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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?

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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.

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