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Let $(x_n)$ be a bounded sequence of real numbers. Define a new sequence $(y_n)$ where $(y_n):=\inf\,(x_k:n\leq\,k)$.

Define $y:= \sup(y_n:n\in\mathbb{N})$.

How to Prove that there exists a subsequence $(x_{n_k})$ of $(x_n)$ such that $\lim(x_{n_k})=y$?

What method should I need to use in this question? I have used contradiction to prove this question but it seems not work.

Could anyone give me some hints to finish the proof or help me to prove this question? Thank you.

Anderson
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  • Check out limsup and liminf and the proof that there is a subsequence converging to each one. ( https://math.stackexchange.com/questions/581128/prove-that-subsequence-converges-to-limsup ) – Jürgen Sukumaran Apr 15 '20 at 19:40
  • @TSF Is that any methods to prove this question without using the theorem and definition of limsup and liminf? It is because I haven't learnt it. – Anderson Apr 15 '20 at 20:03

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