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(en) is ultimately decreasing and that x:= for lim(xn) exists. h Use the fact that the subsequence (x2n) also converges to x to conclude that x = 1. 7. Establish the convergence and find the limits of the following sequences: (a) (1 + 1/n²)*), (6) (1 + 1/#)*). (b) (1+1/2n)"), (d) ((1+2/n)"). 8. Determine the limits of the following. (a) (3n)/2"), (b) ((1+1/2n)"). 9. Suppose that every subsequence of X = (xn) has a subsequence that converges to 0. Show that %3D lim X = 0. %3D 10. Let (xn) be a bounded sequence and for each n E N let s, := sup{xk :k >n} and S:= inf{s,}. Show that there exists a subsequence of (xn) that converges to S. 11. Suppose that xn 20 for all n EN and that lim ((-1)"xn) exists. Show that (xn) converges. 12. Show that if (x,) is unbounded, then there exists a subsequence (xm.) such that lim(1/X) = 0. 13. If xn := (-1)"/n, find the subsequence of (x,) that is constructed in the second proof of the Bolzano-Weierstrass Theorem 3.4.8, when we take I1 := [-1, 1].

(en) is ultimately decreasing and that x:= for lim(xn) exists. h Use the fact that the subsequence (x2n) also converges to x to conclude that x = 1. 7. Establish the convergence and find the limits of the following sequences: (a) (1 + 1/n²)*), (6) (1 + 1/#)*). (b) (1+1/2n)"), (d) ((1+2/n)"). 8. Determine the limits of the following. (a) (3n)/2"), (b) ((1+1/2n)"). 9. Suppose that every subsequence of X = (xn) has a subsequence that converges to 0. Show that %3D lim X = 0. %3D 10. Let (xn) be a bounded sequence and for each n E N let s, := sup{xk :k >n} and S:= inf{s,}. Show that there exists a subsequence of (xn) that converges to S. 11. Suppose that xn 20 for all n EN and that lim ((-1)"xn) exists. Show that (xn) converges. 12. Show that if (x,) is unbounded, then there exists a subsequence (xm.) such that lim(1/X) = 0. 13. If xn := (-1)"/n, find the subsequence of (x,) that is constructed in the second proof of the Bolzano-Weierstrass Theorem 3.4.8, when we take I1 := [-1, 1].

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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Question number (10) only. This material is (Real analysis).
that (xn) is ultimately decreasing and that x := is valid for lim(xn) exists. h Use the fact that the subsequence (x2n) also converges to x to conclude that x = 1. 7 Establish the convergence and find the limits of the following sequences: 1 + 1/#*y*). (6) (1 +1/w)*), (a) (b) (1+1/2n)"), (d) ((1+2/n)"). 8. Determine the limits of the following. (a) ((3n)'/2"), (b) ((1+1/2n)*"). 9. Suppose that every subsequence of X = (xn) has a subsequence that converges to 0. Show that lim X = 0. 10. Let (xn) be a bounded sequence and for each n E N let s, := sup{xk :k >n} and S := inf{s,}. Show that there exists a subsequence of (xn) that converges to S. 11. Suppose that xn 20 for all n EN and that lim((-1)"xn) exists. Show that (xn) converges. 12. Show that if (xn) is unbounded, then there exists a subsequence (Xna) such that lim(1/Xm) = 0. 13. If x := (-1)"/n, find the subsequence of (xn) that is constructed in the second proof of the Bolzano-Weierstrass Theorem 3.4.8, when we take 11 := [-1,1). %3D
Transcribed Image Text:that (xn) is ultimately decreasing and that x := is valid for lim(xn) exists. h Use the fact that the subsequence (x2n) also converges to x to conclude that x = 1. 7 Establish the convergence and find the limits of the following sequences: 1 + 1/#*y*). (6) (1 +1/w)*), (a) (b) (1+1/2n)"), (d) ((1+2/n)"). 8. Determine the limits of the following. (a) ((3n)'/2"), (b) ((1+1/2n)*"). 9. Suppose that every subsequence of X = (xn) has a subsequence that converges to 0. Show that lim X = 0. 10. Let (xn) be a bounded sequence and for each n E N let s, := sup{xk :k >n} and S := inf{s,}. Show that there exists a subsequence of (xn) that converges to S. 11. Suppose that xn 20 for all n EN and that lim((-1)"xn) exists. Show that (xn) converges. 12. Show that if (xn) is unbounded, then there exists a subsequence (Xna) such that lim(1/Xm) = 0. 13. If x := (-1)"/n, find the subsequence of (xn) that is constructed in the second proof of the Bolzano-Weierstrass Theorem 3.4.8, when we take 11 := [-1,1). %3D
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