Let $\displaystyle {x_n \ be \ a \ bounded \ sequence \ ,\ for \ m \ \in N , \ let \ y_m = sup \{x_n ; n \geq m \}} $
$\displaystyle show \ that \ y_m \ is \ convergent $
Let $\displaystyle (x_n)$ be a bounded sequence. For $\displaystyle m \in N$, let $\displaystyle y_m = \sup \{x_n ; n \geq m \}$.
Show that $\displaystyle (y_m)$ is convergent.
Hint: Show that $\displaystyle (y_m)$ is a decreasing sequence that is bounded below.