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

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 $$\displaystyle \lim_{n \rightarrow \infty} y_n = \lim_{n \rightarrow \infty} z_n$$.

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 $$\lim_{n\to +\infty}y_n=\lim_{n\to +\infty}z_n=x$$.