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Let $\{x_n\}_{n \in\mathbb{N}}$ be a bounded sequence of real numbers. Let us define, for each ${n \in\mathbb{N}},$ $$y_n = \sup\{x_k : k \geq n\}$$ and $$z_n = \inf\{x_k : k \geq n\}.$$

Prove that the sequence $\{x_n\}_{n \in\mathbb{N}}$ is convergent if and only if $\displaystyle \lim_{n \rightarrow \infty} y_n = \lim_{n \rightarrow \infty} z_n$.

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First to all $z_n\leq x_n\leq y_n$ for every $n\in\mathbb N$. So if $\lim_{n\to +\infty}z_n=\lim_{n\to +\infty}y_n$ by squeeze theorem also $x_n$ is convergent.

Conversely if $\lim_{n\to +\infty}x_n=x\in\mathbb R$ for every $\varepsilon>0$ there exists $N\in\mathbb N$ so that $x-\varepsilon\leq x_n\leq x+\varepsilon$ for every $N\in\mathbb N$. For every $N'>N$ we have so $$ x-\varepsilon \leq z_{N'}\leq x_n\leq y_{N'}\leq x+\varepsilon $$ for every $n\geq N'$, in other words $\lim_{n\to +\infty}y_n=\lim_{n\to +\infty}z_n=x$.

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