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So I'm given a decreasing family of compact sets $(K_n)$ in $\mathbb{R}$ such that $K_1\supset K_2\supset K_3\supset \cdots$ and have to show that for a sequence $(a_n)$ such that $a_n \in K_n$ there exists a subsequence $(a_{n_k})$ that converges to a point $a \in \bigcap_{n \in \mathbb{N}}{K_n}$.

I am able to show that this intersection is nonempty but have no idea as to how to approach that problem of showing that there exists a convergent subsequence? Thanks in advance!

EDIT: Ok, so I was able to prove that it has a convergent sequence but now how would I go about showing that the value this sequence converges to is in the intersection?

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Let $n \in \mathbb{N}$. The tail $a_n,a_{n+1},\dots$ is contained in $K_n$, which is a compact set, and in particular is a closed set. So the tail of $a_{n_k}$ which is contained in this tail converges to $a$ and is contained in the closed set $K_n$. So $a \in K_n$. Since this $n$ was arbitrary, $a \in \bigcap_{n=1}^\infty K_n$.

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