Proving Upper and Lower Bounds for Continuous Happiness on a Scale of -10 to 10

[Numnum]

Homework Statement

Assume happiness can be measured on a scale from -10 (very unhappy) to 10 (extremely happy). Let D denote the set of possible ways to live one day. For each way of living a day$$d∈D$$ define a function
$h_d$$$:[0,24]⟶[−10,10]$$ to be equal to the value of happiness at each point in time. For example, $h_d$$$(6.25)=−7.3$$ means that for this particular way of living a day the level of happiness at 6:15 in the morning is -7.3 (this may happen if one sets alarm clock to 6:00am). Let $H_d$=glb {$h_d$(t)|{0≤t≤24}. Let H=lub{$H_d$|d∈D}.

1) Assuming that happiness changes continuously, prove that for every d∈D there exists t∈ [0,24], such that $h_d$ (t)= $H_d$.

2) Assume that happiness changes continuously and that H=4. Is it possible to live through a day so that you are always happier than 3.9999999? Is it possible to live through a day so that you are always happier than 4? Justify your answers.

3) Without assuming that happiness changes continuously, prove that {−10≤H≤10}.

Homework Equations

The Attempt at a Solution

Can you walk me through what the question is asking? I've been having difficulty with understanding what exactly is the glb and lub.

 

[tiny-tim]

Hi Numnum!

Each h_d is a function, and you can draw it on a 24-hour graph.

The minimum on that particular graph is H_d.

The greatest H_d (for all d in D) is H.

 

[Numnum]

So for part a) I have to prove that for every $h_d$ function, there exists a minimum? I'm not very good at proofs, so... What would I choose as the function?

 

[tiny-tim]

Hi Numnum!

(just got up :zzz:)

What would I choose as the function?

I don't understand.

 

[Ray Vickson]

Homework Statement

Assume happiness can be measured on a scale from -10 (very unhappy) to 10 (extremely happy). Let D denote the set of possible ways to live one day. For each way of living a day$$d∈D$$ define a function
$h_d$$$:[0,24]⟶[−10,10]$$ to be equal to the value of happiness at each point in time. For example, $h_d$$$(6.25)=−7.3$$ means that for this particular way of living a day the level of happiness at 6:15 in the morning is -7.3 (this may happen if one sets alarm clock to 6:00am). Let $H_d$=glb {$h_d$(t)|{0≤t≤24}. Let H=lub{$H_d$|d∈D}.

1) Assuming that happiness changes continuously, prove that for every d∈D there exists t∈ [0,24], such that $h_d$ (t)= $H_d$.

2) Assume that happiness changes continuously and that H=4. Is it possible to live through a day so that you are always happier than 3.9999999? Is it possible to live through a day so that you are always happier than 4? Justify your answers.

3) Without assuming that happiness changes continuously, prove that {−10≤H≤10}.

Homework Equations

The Attempt at a Solution

Can you walk me through what the question is asking? I've been having difficulty with understanding what exactly is the glb and lub.

1) is asking you to prove a very standard result, viz., that a continuous function f on a finite closed interval [a,b] attains a maximum and a minimum.