- Complete The Proof Of The Theorem Below 1 (20.27 KiB) Viewed 23 times
Complete the Proof of the theorem below
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Complete the Proof of the theorem below
Complete the Proof of the theorem below
1 y-X Theorem 2.10 If x and y are real numbers with x <y, then there exists a rational number r with x <r<y. Proof: We only look at the case where x >0. Since x < y, we have y - x > 0. By the Archimedean Principle, we can find n € N such that n> Hence, n(y - x) > 1 and we get ny > nx+1. Using a corollary to the Archimedean Principle, take m EN such that m-15 nx <m. Observe that m s nx +1. Together with the inequality above, we now get nx <m s nx +1 <ny. Thus, nx <m<ny giving us x<<y. m n
1 y-X Theorem 2.10 If x and y are real numbers with x <y, then there exists a rational number r with x <r<y. Proof: We only look at the case where x >0. Since x < y, we have y - x > 0. By the Archimedean Principle, we can find n € N such that n> Hence, n(y - x) > 1 and we get ny > nx+1. Using a corollary to the Archimedean Principle, take m EN such that m-15 nx <m. Observe that m s nx +1. Together with the inequality above, we now get nx <m s nx +1 <ny. Thus, nx <m<ny giving us x<<y. m n
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