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HW3: Prove that U=(x| x>1} is an open set on the real line. Hint: Start the proof by drawing the set U on the line, marking a point p in U and guessing a radius that will work. The r_p you guess will be a formula depending on the real number p. For example: when p-4, you could take r_p=3 because (4-3, 4+3)=(1,7) is a subset of U=(1,\infty). Here linfty is tex for infinity. What formula do you think would work for any p? Check your formula for p=2, p35, and p%3D100. Then start the proof in the following format as in the video where you fill in the formula for the radius r_p you guessed. 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 whyr_p>0 here

HW3: Prove that U=(x| x>1} is an open set on the real line. Hint: Start the proof by drawing the set U on the line, marking a point p in U and guessing a radius that will work. The r_p you guess will be a formula depending on the real number p. For example: when p-4, you could take r_p=3 because (4-3, 4+3)=(1,7) is a subset of U=(1,\infty). Here linfty is tex for infinity. What formula do you think would work for any p? Check your formula for p=2, p35, and p%3D100. Then start the proof in the following format as in the video where you fill in the formula for the radius r_p you guessed. 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 whyr_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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Defn: A set U in a metric space X is an open set if for every p in U there is a radius r_p >0 such that B(p, r_p) is a subset of U. Here r_p is the tex notation for r subscript p. Write this in your notes with quantifier symbols replacing "for every" and "there is". A set U is an open set if Vpeu 3rp> O sit B(p,rp)c U. p depends on P be uery small can , can be large HW3: Prove that U={x| x>1} is an open set on the real line. Hint: Start the proof by drawing the set U on the line, marking a point p in U and guessing a radius that will work. The r_p you guess will be a formula depending on the real number p. For example: when p-4, you could take r_p=3 because (4-3, 4+3)=(1,7) is a subset of U=(1,\infty). Here linfty is tex for infinity. What formula do you think would work for any p? Check your formula for p=2, p=5, and p=100. Then start the proof in the following format as in the video where you fill in the formula for the radius r_p you guessed. 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. 1<p-r_p<x (this works if you chose the right formula for r_p) >0 (1) explain why r_p>0 here 5. x in U Notice that in Steps 2-5 you are proving the ball B(p,r_p) is a subset of U. If you need more than 5 steps then take more steps. It depends how complicated your formula for r_p is.
Transcribed Image Text:Defn: A set U in a metric space X is an open set if for every p in U there is a radius r_p >0 such that B(p, r_p) is a subset of U. Here r_p is the tex notation for r subscript p. Write this in your notes with quantifier symbols replacing "for every" and "there is". A set U is an open set if Vpeu 3rp> O sit B(p,rp)c U. p depends on P be uery small can , can be large HW3: Prove that U={x| x>1} is an open set on the real line. Hint: Start the proof by drawing the set U on the line, marking a point p in U and guessing a radius that will work. The r_p you guess will be a formula depending on the real number p. For example: when p-4, you could take r_p=3 because (4-3, 4+3)=(1,7) is a subset of U=(1,\infty). Here linfty is tex for infinity. What formula do you think would work for any p? Check your formula for p=2, p=5, and p=100. Then start the proof in the following format as in the video where you fill in the formula for the radius r_p you guessed. 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. 1<p-r_p<x (this works if you chose the right formula for r_p) >0 (1) explain why r_p>0 here 5. x in U Notice that in Steps 2-5 you are proving the ball B(p,r_p) is a subset of U. If you need more than 5 steps then take more steps. It depends how complicated your formula for r_p is.
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