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We assume that x0. Therefore there exists a natural number n such that n> Since z is fixed we can simply evaluate 1 and take the next largest natural number. Which of the following completes the proof?

We assume that x0. Therefore there exists a natural number n such that n> Since z is fixed we can simply evaluate 1 and take the next largest natural number. Which of the following completes the proof?

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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Why is the one I chose wrong? Which is the correct one and why?

Therefore there must exist a natural number m such that nx < m< ny, and by dividing through by n we have x< m <y. Since n and m are both natural numbers and hence also integers, we have found a rational number existing between x and y, as required. We have that n>1 = nz <1 = ny-nx < 1 Therefore there must exist a natural number m such that ny - nx <m, and hence y-x< m so y <x+ m, This shows that x+ m is a rational number greater than x but less than y. Since n and m are both natural numbers and hence also integers, we have found a rational number existing between x and y, as required. O None of these. We have that Therefore x <1<y so 1 is a rational number existing between x and y as required. We have that y-x n(y-x) Therefore there is a gap greater than 1 between y and x so it must necessarily contain an integer, which is a rational number. Therefore we have found a rational number between x and y, as required.
Transcribed Image Text:Therefore there must exist a natural number m such that nx < m< ny, and by dividing through by n we have x< m <y. Since n and m are both natural numbers and hence also integers, we have found a rational number existing between x and y, as required. We have that n>1 = nz <1 = ny-nx < 1 Therefore there must exist a natural number m such that ny - nx <m, and hence y-x< m so y <x+ m, This shows that x+ m is a rational number greater than x but less than y. Since n and m are both natural numbers and hence also integers, we have found a rational number existing between x and y, as required. O None of these. We have that Therefore x <1<y so 1 is a rational number existing between x and y as required. We have that y-x n(y-x) Therefore there is a gap greater than 1 between y and x so it must necessarily contain an integer, which is a rational number. Therefore we have found a rational number between x and y, as required.
We assume that x<y. We can make this assumption without loss of generality, since we can make this statement true by simply relabeling. Since x<y, there exists another real number z such that y-x = z and z>0. Therefore there exists a natural number n such that n>1 Since z is fixed we can simply evaluate 1 and take the next largest natural number. Which of the following completes the proof? We have that n>1 = nz >1 a ny-nx> 1. Therefore there must exist a natural number m such that nx < m < ny, and by dividing through by n we have x m <y. Sincen and m are both natural numbers and hence also integers, we have found a rational number existing between x and y, as required. We have that n> » nz <1 → ny-nx < 1 Therefore there must exist a natural numberm such that ny - nx <m, and hence y - x < so y <x + . This shows that x+ is a rational number greater than x but less than y. Since n and m are both natural numbers and hence also Integers, we have found a rational number existing between x and y, as required. O None of these. We have that - y-x> n Therefore x <<y so is a rational number existing between x and y as required. 90 1155 We have that
Transcribed Image Text:We assume that x<y. We can make this assumption without loss of generality, since we can make this statement true by simply relabeling. Since x<y, there exists another real number z such that y-x = z and z>0. Therefore there exists a natural number n such that n>1 Since z is fixed we can simply evaluate 1 and take the next largest natural number. Which of the following completes the proof? We have that n>1 = nz >1 a ny-nx> 1. Therefore there must exist a natural number m such that nx < m < ny, and by dividing through by n we have x m <y. Sincen and m are both natural numbers and hence also integers, we have found a rational number existing between x and y, as required. We have that n> » nz <1 → ny-nx < 1 Therefore there must exist a natural numberm such that ny - nx <m, and hence y - x < so y <x + . This shows that x+ is a rational number greater than x but less than y. Since n and m are both natural numbers and hence also Integers, we have found a rational number existing between x and y, as required. O None of these. We have that - y-x> n Therefore x <<y so is a rational number existing between x and y as required. 90 1155 We have that
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