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HW4: Prove that U={x| x<4} is an open set on the real line. Hint: If you have a hard time guessing the value for r_p that works for p=3, p=-2, and p=- 100 that is alright. You can still complete the proof. Write out all the steps with the variable r_p as follows: 1. Given any p in U we chooser_p= 2. Given any x in B(p,r_p) we have x in (p-r_p, p+r_p) (2) B(p,R)=(p-R,p+R) 3. p-r_p0 (1) explain why r_p>0 here

HW4: Prove that U={x| x<4} is an open set on the real line. Hint: If you have a hard time guessing the value for r_p that works for p=3, p=-2, and p=- 100 that is alright. You can still complete the proof. Write out all the steps with the variable r_p as follows: 1. Given any p in U we chooser_p= 2. Given any x in B(p,r_p) we have x in (p-r_p, p+r_p) (2) B(p,R)=(p-R,p+R) 3. p-r_p0 (1) explain why r_p>0 here

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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HW5: Prove that the interval U=(5,9) is an open set on the real line. Hint: You will need to choose an r_p small enough so that your ball stays under 9 and above 5. To do this you will taking a minimum of two different formulas. This is similar to the minimum used in the video proving intersections of open sets are open. Again you may have difficulty guessing a formula, so instead start writing the proof and figure it out from the bottom up: 1. Given any p in U we choose r_p=_ 2. Given any x in B(p,r_p) we have x in (p-r_p, p+r_p) (2) B(p,R)=(p-R,p+R) 3. p-r_p<x<p+r_p 4. 5< p-r_p<x<p+r_p <9_solve for r_p so that this line works 5. x in U This time r_p has two inequalities to solve: 5< p-r_p AND p+r_p <9 So after solving you will see that you need r_p<p-5 AND r_p< 9-p So at the top of the proof you choose r_p = min{p-5, 9-p}>0 and justify why this is greater than zero carefully. And then complete your proof downward. >0 (1) explain why r_p>0 here
Transcribed Image Text:HW5: Prove that the interval U=(5,9) is an open set on the real line. Hint: You will need to choose an r_p small enough so that your ball stays under 9 and above 5. To do this you will taking a minimum of two different formulas. This is similar to the minimum used in the video proving intersections of open sets are open. Again you may have difficulty guessing a formula, so instead start writing the proof and figure it out from the bottom up: 1. Given any p in U we choose r_p=_ 2. Given any x in B(p,r_p) we have x in (p-r_p, p+r_p) (2) B(p,R)=(p-R,p+R) 3. p-r_p<x<p+r_p 4. 5< p-r_p<x<p+r_p <9_solve for r_p so that this line works 5. x in U This time r_p has two inequalities to solve: 5< p-r_p AND p+r_p <9 So after solving you will see that you need r_p<p-5 AND r_p< 9-p So at the top of the proof you choose r_p = min{p-5, 9-p}>0 and justify why this is greater than zero carefully. And then complete your proof downward. >0 (1) explain why r_p>0 here
HW4: Prove that U={x| x<4} is an open set on the real line. Hint: If you have a hard time guessing the value for r_p that works for p=3, p=-2, and p=- 100 that is alright. You can still complete the proof. Write out all the steps with the variable r_p as follows: 1. Given any p in U we chooser_p=. 2. Given any x in B(p,r_p) we have x in (p-r_p, p+r_p) (2) B(p,R)=(p-R,p+R) 3. p-r_p<x<p+r_p 4. x<p+r_p<4 solve for r_p so that this line works 5. x in U So now solve for the r_p that will work to give (4) and plug it in on the top and complete your proof. _>0 (1) explain why r_p>0 here
Transcribed Image Text:HW4: Prove that U={x| x<4} is an open set on the real line. Hint: If you have a hard time guessing the value for r_p that works for p=3, p=-2, and p=- 100 that is alright. You can still complete the proof. Write out all the steps with the variable r_p as follows: 1. Given any p in U we chooser_p=. 2. Given any x in B(p,r_p) we have x in (p-r_p, p+r_p) (2) B(p,R)=(p-R,p+R) 3. p-r_p<x<p+r_p 4. x<p+r_p<4 solve for r_p so that this line works 5. x in U So now solve for the r_p that will work to give (4) and plug it in on the top and complete your proof. _>0 (1) explain why r_p>0 here
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