1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.