supremum and infinum

**novice** · September 11th 2010, 08:11 PM

Theorem

$E \subseteq \mathbb{R}, E \not = \varnothing$

sup $E$ exists $\Leftrightarrow$ inf $(-E)$ exists

Proof:

$"\Rightarrow"$

$supE$ exists

Now we show that $-supE=inf (-E).$
Show that
1. $-sup E$ is a lower bound of $-E$ .
2. if $s$ is a lower bound of $-E \Rightarrow s \leq$ -sup $E.$

1.
$\because$ $sup E$ is an upper bound of $E$
$\therefore x \leq$ $sup E, \forall \in E \Rightarrow -x \geq$ $-sup E, \forall x \in E$

$\therefore -sup E$ is a lower bound of $-E$ (line added for correction 6:30 a.m. Sep 12)

2. Suppose that $s$ is a lower bound of $-E$

suppose not $\Rightarrow s >$ $-sup E \Rightarrow -s <$ $sup E$

on the other hand

$-x \geq s, \forall x \in E$ <------(Could someone tell me how this come about.)

$\therefore x \leq -s$

Hence, $-s$ is an upper bound of $E \rightarrow\leftarrow$

By 1. & 2, $inf (-E) \exists$ & $inf (-E)$ = $sup E$

**Plato** · September 12th 2010, 03:11 AM

I don’t follow what you posted.
Suppose that $s = \sup (E)$ then if $y\in -E$ then $-y\in E$ .
This means that $-y\le s$ or $y \ge -s$ .
This shows that $- s \le \inf ( - E)$ .
Suppose that $- s < \inf ( - E)$ then $s>-\inf(-E)$ .
That means $\left( {\exists z \in E} \right)\left[ { - \inf ( - E) < z \le s} \right]$ or $-z<\inf(-E)$ .
What is wrong with that?

**novice** · September 12th 2010, 10:15 AM

Originally Posted by Plato

I don’t follow what you posted.
Suppose that $s = \sup (E)$ then if $y\in -E$ then $-y\in E$ .
This means that $-y\le s$ or $y \ge -s$ .
This shows that $- s \le \inf ( - E)$ . <----This part makes a whole world of a difference.
Suppose that $- s < \inf ( - E)$ then $s>-\inf(-E)$ .
That means $\left( {\exists z \in E} \right)\left[ { - \inf ( - E) < z \le s} \right]$ or $-z<\inf(-E)$ .
What is wrong with that?

My original post was from lecture notes of some school of mathematics I found on the web--I don't remember where I got it. There is a missing link in the proof which made it hard to understand.

Since you said you could not follow it, you made me feel better.

Your instructions are superb, which I am very glad that you are on board.
I am not a math major, so I don't have access to a real math professor, but thanks to MHF for giving me access to a teacher like you.

Thank you, sir.

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