MHB Analysis question involving Supremums and Infimums.

[pineapplechem]

Hi, first post - need some hints/help with the question attached, please. I have no idea where to go with it, to be honest.

 

[Evgeny.Makarov]

Hi, and welcome to the forum.

Here is what I made out from the picture.

Given a bounded sequence $(a_n)_{n\in\Bbb N}$, define $A_k\subset\Bbb R$ by $A_k:=\{a_n:n\ge k\}$ and set $a_k=\sup A_k$ and $b_k=\inf A_k$. Explain why $(a_k)_{k\in\Bbb N}$ is a decreasing sequence and why $(b_k)_{k\in\Bbb N}$ is an increasing sequence.

I renamed $B_k$ to $b_k$: this way uppercase $A_k$ denote sets and lowercase $a_k$ and $b_k$ denote numbers. This still seems wrong because both the original sequence and the defined one are denote by $a_k$. Could you click "Reply with Quote" button, examine the code that produces formulas and correct the problem statement as necessary?

 

[pineapplechem]

Hi, and welcome to the forum.

Here is what I made out from the picture.

Given a bounded sequence $(a_n)_{n\in\Bbb N}$, define $A_k\subset\Bbb R$ by $A_k:=\{a_n:n\ge k\}$ and set $a_k=\sup A_k$ and $b_k=\inf A_k$. Explain why $(a_k)_{k\in\Bbb N}$ is a decreasing sequence and why $(b_k)_{k\in\Bbb N}$ is an increasing sequence.

I renamed $B_k$ to $b_k$: this way uppercase $A_k$ denote sets and lowercase $a_k$ and $b_k$ denote numbers. This still seems wrong because both the original sequence and the defined one are denote by $a_k$. Could you click "Reply with Quote" button, examine the code that produces formulas and correct the problem statement as necessary?

You've interpreted it correctly - however what you changed to lower case b is meant to be Beta, and the one before it that looks like an a is meant to be alpha, I just didn't write them very well. Sorry about the confusion but it's now correct. Ak is defined for each k in the positive natural numbers as well.

 

[Evgeny.Makarov]

So $\alpha_1=\sup\{a_1,a_2,a_3,\dots\}$ and $\alpha_2=\sup\{a_2,a_3,\dots\}$. Which one is bigger in general?

Formally, $\gamma=\sup C$, by definition, if two properties hold:

(1) $\gamma\ge x$ for all $x\in C$ (this means that $\gamma$ is an upper bound), and
(2) if $\delta\ge x$ for all $x\in C$, then $\gamma\le\delta$ (this means that $\gamma$ is the smallest possible upper bound).

Here we have $\alpha_1\ge x$ for all $x\in A_1$ by (1), and since $A_2\subseteq A_1$, it follows that $\alpha_1\ge x$ for all $x\in A_2$. We also know that $\alpha_2=\sup A_2$. What does property (2) say when applied to $\delta=\alpha_1$, $\gamma=\alpha_2$ and $C=A_2$?