Suppose that $\displaystyle \displaystyle{x_n}$ is a bounded sequence of real numbers.Show that $\displaystyle \displaystyle{y_n=\sup_{m \geq n}x_m}$ is a bounded decreasing sequence and deduce that $\displaystyle \displaystyle{y_n \longrightarrow y}$,ie $\displaystyle \displaystyle{\limsup_{n\to\infty}x_n=y}$.