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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please transcribe the text in digital format so that you can copy it to microsoft word, thanks
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec163e48-c9dd-4e6d-808e-8805f766c93c%2F135f450d-d1a5-4e8b-bfa3-a81584d744e4%2Fbu18wf8_processed.jpeg&w=3840&q=75)
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.
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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