Suppose that $(x_n)$ and $(y_n)$ are sequences and $x_n^2+y_n^2=1$ for all $n\in \mathbb{N}$ .Show that there are positive integers $n_1,n_2,n_3,...$ with $n_1<n_2<n_3...$ such that both $(x_{n_k})$ and $(y_{n_k})$ are converges

how to prove from all sides i am stuck

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    $\begingroup$ What can you use, do you know about compact sets? The (Bernand) Bolzano-(Karl) Weierstraß theorem on limit points of bounded sequences? $\endgroup$ – Lutz Lehmann Nov 1 '17 at 11:50

Define the sequence $(z_n = (x_n,y_n))_{n\in\mathbb{N}}$ on $\mathbb{R}^2$. Then $z_n$ lies on the unit circle for all $n\in\mathbb{N}$, which is a compact set. Therefore there exists a converging subsequence.

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