Convergence of sequence of sups and sequence of infs of bounded sequence

Question

Let $\{x_n\}$ be a bounded sequence of real numbers. Let us define $y_n := \text{sup}\{x_k : k \ge n\}$ and $z_n := \text{inf}\{x_k : k \ge n\}$. Deduce that $\{y_n\}$ and $\{z_n\}$ are convergent sequences.

Answer 1

Check out limsup and liminf. One way of the defining them is precisely as the limit of your $y_n$ and $z_n$ sequences. An important property is that they both always exist (even in the case $x_n$ is unbounded as long as you allow limits to $\pm \infty$).

Try to prove that $y_n, z_n$ are monotone, bounded sequences. A standard result then gives that they are convergent.