made fact from lecture 13 (somewhat) less trivial
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@ -274,7 +274,8 @@ We have shown, that $\mu_{n_k} \implies \mu$ along a subsequence.
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We still need to show that $\mu_n \implies \mu$.
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We still need to show that $\mu_n \implies \mu$.
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\begin{fact}
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\begin{fact}
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Suppose $a_n$ is a bounded sequence in $\R$,
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Suppose $a_n$ is a bounded sequence in $\R$,
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such that any subsequence converges to $a \in \R$.
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such that any subsequence has a subsequence
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that converges to $a \in \R$.
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Then $a_n \to a$.
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Then $a_n \to a$.
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\end{fact}
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\end{fact}
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\begin{subproof}
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\begin{subproof}
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