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Liven, 18 Q sequuence is cauchy Hhen we have to Show that any Subseence of the Sequenceanealso a Couchy Seuence. Proof2. Above Stalement is Proved by smg the tolloung two Stalemente O A Saquene

Liven, 18 Q sequuence is cauchy Hhen we have to Show that any Subseence of the Sequenceanealso a Couchy Seuence. Proof2. Above Stalement is Proved by smg the tolloung two Stalemente O A Saquene

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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Since e is a limit Point oft <Sn), thene must exist a POSHIVR in teger Aiso K>m. Thene fone from cU we have Smar - 2 =| Smap - Sm t Sm - Sx tSx -e] Hence the sequence Sn) Convenges to l. (T) - Every Len Subsequence of a convengent Seauence is convergent an d Convenges to the Same number Proof- then au Subsequen ces If Convenges to e then {sn) is bounded of (Sn) will be bounded. So by the Bol Zano wein Stnaes theomem, ae Subsequeencec of <Sn) has a limit Point but eveny limito Pont Of A Subsequences is also a limit Pornt of (Sn) . So eveny Sulsequence Ot (3n) is converngent. we ol neady moved that eveniy Convengent sequence is Cauchy of Seuence : Hence eveny Subsequence convengent Sequeence which s GSo a lauchy searen ce is convengent thaet iseauchy, keta Eveny Subsequence of Cauchy Seruence is also a Cauchy (Provedl). Segueence.
Transcribed Image Text:Since e is a limit Point oft <Sn), thene must exist a POSHIVR in teger Aiso K>m. Thene fone from cU we have Smar - 2 =| Smap - Sm t Sm - Sx tSx -e] Hence the sequence Sn) Convenges to l. (T) - Every Len Subsequence of a convengent Seauence is convergent an d Convenges to the Same number Proof- then au Subsequen ces If Convenges to e then {sn) is bounded of (Sn) will be bounded. So by the Bol Zano wein Stnaes theomem, ae Subsequeencec of <Sn) has a limit Point but eveny limito Pont Of A Subsequences is also a limit Pornt of (Sn) . So eveny Sulsequence Ot (3n) is converngent. we ol neady moved that eveniy Convengent sequence is Cauchy of Seuence : Hence eveny Subsequence convengent Sequeence which s GSo a lauchy searen ce is convengent thaet iseauchy, keta Eveny Subsequence of Cauchy Seruence is also a Cauchy (Provedl). Segueence.
Liven, 18 Q Sequence is cauchy then we have to Show that æny Subseuence of the Sequencearnealso a Couchy Seueence. Proof2. Above Stalement is Proved by smg the tolloung two Stalemente O A Sequenee <on> Convenges it and oniy it it a Cauchy 8ewence Proof Neces sony CondiHon Let the sequeence (sn be convengent wowch Converges to l. Then ton a given E7o, thereexist a Positive integen m Such that jpno wzuz dtu wont ik d 1 + = G b>m and P> ! This- Shows that Sn) is cauchy Sequence Sufficient Condition Let <Sn) be a coare chy sequence. Then <sn) is boundled By B0lzano weines fnass theonen <Sn> bas al least One Lmit Pont say e. ue Shal Show that the Sequeen ce (Sn) Con venges to l By the ven Condition., we have thal fon a giren E >o theore exist a positive integer Such that + nam and P 1 In Penticulan fon n-m we heive
Transcribed Image Text:Liven, 18 Q Sequence is cauchy then we have to Show that æny Subseuence of the Sequencearnealso a Couchy Seueence. Proof2. Above Stalement is Proved by smg the tolloung two Stalemente O A Sequenee <on> Convenges it and oniy it it a Cauchy 8ewence Proof Neces sony CondiHon Let the sequeence (sn be convengent wowch Converges to l. Then ton a given E7o, thereexist a Positive integen m Such that jpno wzuz dtu wont ik d 1 + = G b>m and P> ! This- Shows that Sn) is cauchy Sequence Sufficient Condition Let <Sn) be a coare chy sequence. Then <sn) is boundled By B0lzano weines fnass theonen <Sn> bas al least One Lmit Pont say e. ue Shal Show that the Sequeen ce (Sn) Con venges to l By the ven Condition., we have thal fon a giren E >o theore exist a positive integer Such that + nam and P 1 In Penticulan fon n-m we heive
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