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 $C=A_2$?

 

[blue_raver22]

Sure, I'd be happy to help you with your analysis question involving supremums and infimums. Can you provide more context or information about the question? What specific concepts or topics are being discussed? This will help me provide a more tailored response. Additionally, have you tried looking at any relevant equations or definitions related to supremums and infimums? Understanding these concepts is crucial for solving problems involving them. Let me know and I'll do my best to guide you in the right direction.