Posts Tagged ‘photos’

Set Greatest

Set Greatest

If a set A of real numbers has both a least upper bound and a greatest lower bound, show that inf A <= sup A?

If a set A of real numbers has both a least upper bound and a greatest lower bound, show that inf A <= sup A

The hypothesis that A has a least upper bound and a greatest lower bound implies that A is nonempty. (If A is empty, any real number is both an upper bound and a lower bound for A, so there is no least upper bound or greatest lower bound.)

Since A is nonempty there is a real number x satisfying x in A.

Since inf A is a lower bound for A and x is in A, inf A <= x.
[This inference is made just from the definition of what it means for a number to be a lower bound for a set.]

Since sup A is an upper bound for A and x is in A, sup A <= x.
[This inference is made just from the definition of what it means for a number to be an upper bound for a set.]

Since inf A <= x and x <= sup A we conclude that inf A <= sup A. [This inference comes from a property of <= on real numbers: if r <= s and s <= t, then r <= t.]

The same proof shows that if L is _any_ lower bound for A, and R is _any_ upper bound for for A, then L <= R. We do not need to use the fact that inf A is the greatest lower bound or that sup A is the least upper bound.

Greatest Skateboarding Tricks

Minogue Kylie

Minogue Kylie

I have had strange dreams about Kylie Minogue since I was small. Do you get those dreams?

I mean dreams of celebrities. I hope she gets the message if she uses the internet and it would be a privilege to meet her some day and that she knows me by the starchild Aj the Dj.

I need to experience events with her and do a march like disco inferno.

I don't care how long it takes. My heart bleeds for her.

F**k me sounds more like a night mare to me!!
think you need to get to hospital straight away!!!
As a nurse got to say if your heart is bleeding it is really not a good thing to be happening to you? can you answer me a question purely for medical research?? did your heart bleed before or after you did the funky disco inferno or after... sounds to me like that a dangerous pass time??
were you spinning the discs while rocking out to rock dj and secretly doing a robbie williams ? was she screaming aj aj give me some inxs ??

Kylie Minogue - All The Lovers

Red Album

Red Album

How is The Games "RED Album" coming along?

Does he have any singles lined up,
album drop dates,
anything important?
featured artists

I've heard a few songs on it, and the were great. No singles have been released yet. It should drop around December 09. Featured artist will be, Lil Wayne, Lady GaGa, Pharrell, Kanye West and Gucci Mane.

1500 TV : The making of The Game's "THE RED ALBUM" Episode 2

Last Call

Last Call

Lee Ann Womack - Last Call

View From

View From

Hot Planet - Cold Comfort