• November 18th 2009, 11:16 AM
Natedogg5106
I'm having a lot of trouble with contractions and open coverings for my real analysis class, and I was hoping someone would be able to help me with a few of the problems:

1) Give an example of a continuous bounded function on (-inf, inf) that attains neither a maximum nor minimum value

2a) For all x in (0,1), let I_x denote the open interval (x/2, (x+1)/2). Show that the family G of all such I_x is an open covering of (0,1) which admits no finite subcovering of (0,1)
2b) Add two appropriate sets to the family G (from above) to form an open covering H of [0,1]. Show that H does admit a finite subcovering of [0,1].
• November 18th 2009, 11:57 AM
Plato
Quote:

Originally Posted by Natedogg5106
1) Give an example of a continuous bounded function on (-inf, inf) that attains neither a maximum nor minimum value

Do you know about $\arctan(x)?$
• November 18th 2009, 12:13 PM
Drexel28
Quote:

Originally Posted by Natedogg5106
2a) For all x in (0,1), let I_x denote the open interval (x/2, (x+1)/2). Show that the family G of all such I_x is an open covering of (0,1) which admits no finite subcovering of (0,1)
2b) Add two appropriate sets to the family G (from above) to form an open covering H of [0,1]. Show that H does admit a finite subcovering of [0,1].

2a) So you are attempting to show that $\mathcal{I}=\left\{I_x=\left(\frac{x}{2},\frac{x+1 }{2}\right):x\in(0,1)\right\}$ is an open cover of $(0,1)$. Did you try it? Did you show that $(0,1)\subseteq \bigcup_{I_x\in\mathcal{I}}I_x$? What happens if you take one of the elmements out?

2b). Hint: When you add the two appropriate sets you only need three intervals to cover it.
• November 19th 2009, 05:24 PM
Natedogg5106
Alright thank you very much. I didn't even think to use any inverse trigonometric functions. I think I figured out the second problem, although the two sets I added probably aren't the right ones, it seems to work to me. How do I give you guys feedback or whatnot? This is my first time using this forum
• November 19th 2009, 05:31 PM
Drexel28
Quote:

Originally Posted by Natedogg5106
Alright thank you very much. I didn't even think to use any inverse trigonometric functions. I think I figured out the second problem, although the two sets I added probably aren't the right ones, it seems to work to me. How do I give you guys feedback or whatnot? This is my first time using this forum

Either press the thank button, which you did, or press the gree plus box where it says "rep".