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Let $a_n$ be a bounded sequence. Assume that the only subsequent limits of $a_n$ are $\pm1$. Prove that $\lim\limits_{n\to\infty}|a_n|=1$.
2026-04-12 22:46:26.1776033986
limits of partial sequences
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Since $|\cdot|$ is continuous, $\liminf a_n=-1$ and $\limsup a_n=1$ ($a_n$ is bounded) we have $$1=\liminf_{n\rightarrow\infty}|a_n|$$ and $$1=\limsup_{n\rightarrow\infty}|a_n|.$$ Therefore $\lim |a_n|=1$.