# How to show that the following sequence is bounded and decreasing?

Suppose we have a bounded sequence $a_i$ and another sequence such that for each $j \in \mathbb N$ we have $b_j = \mathrm {sup} \{a_i: \ i \ge j \}$. How to show that $b_j$ is a bounded and decreasing sequence?

I presume that I can use induction to show that the sequence is bounded and decreasing but I do not know how to set the base cases and the inductive hypotheses.

the number of elements of the set you take the upper bound from is getting fewer elements with each increase of $j$. so the sequence of upper bounds will not increase, it can only decrease. (note that it may be constant if $\left(a_{n}\right)_{n\in\mathbb{N}}$ is constant)