Proving convergent sequences are Cauchy sequences












4












$begingroup$



Prove that if $x_n rightarrow a, n rightarrow infty$ then ${x_n}$ is a Cauchy sequence.




I believe I have found the proof as follows, wondering if there are any simpler methods or added intuition. For me, it makes sense that if a sequence has a limit, then distances between elements in the sequence must be getting smaller, in order for it to converge.



Given $epsilon > 0, exists N_1 s.t. forall n geq N_1:$



$|x_n - a| < frac{epsilon}{2} < epsilon$



and for $m > n geq N_1$ we also have:



$|x_m -a| < frac{epsilon}{2} < epsilon$



Let $N geq N_1$, then $forall n,m geq N$ we have:



$|x_n-x_m| = |x_n - a -x_m+a| < |x_n - a| + |-(x_m -a)| < frac{epsilon}{2} + frac{epsilon}{2} = epsilon$



Therefore ${x_n}$ is a Cauchy sequence.



Also, if a sequence is Cauchy does it always converge? In other words, is it sufficient to check if a sequence is Cauchy to check for convergence.






real-analysis limits proof-verification cauchy-sequences







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    "if a sequence is not Cauchy can it converge?" you literally just proved if it converges it is Cauchy....
    $endgroup$
    – mathworker21
    Jan 29 '17 at 23:17






  • 4




    $begingroup$
    Very good proof. Indeed, if a sequence is convergent, then it is Cauchy (it can't be not Cauchy, you have just proved that!). However, the converse is not true: A space where all Cauchy sequences are convergent, is called a complete space. In complete spaces, Cauchy property is equivalent to convergence. However (for example Riemann integrable functions on $mathbb R$, say) incomplete spaces certainly do exist, but there is an abstract notion of "completion" of a space, which is used time and again to unify these concepts.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Jan 29 '17 at 23:18












  • $begingroup$
    Yes haha sorry, what I meant to ask was if a sequence is Cauchy does it always converge, which астон вілла олоф мэллбэрг explained nicely!
    $endgroup$
    – student_t
    Jan 29 '17 at 23:24












  • $begingroup$
    It is true that a Cauchy sequence in $Bbb R$ converges to a member of $Bbb R.$ This is a consequence of the definition of $Bbb R.$ A Cauchy sequence in $Bbb Q$ will converge in $Bbb R$ but it may or may not converge to a member of $Bbb Q$. So whether a sequence is said to converge depends on the $space$ (e.g. $Bbb Q$ or $Bbb R$ ) that is under consideration.
    $endgroup$
    – DanielWainfleet
    Dec 17 '17 at 5:53










  • $begingroup$
    the title is opposed to your question
    $endgroup$
    – Guy Fsone
    Jan 16 '18 at 12:51
















4












$begingroup$



Prove that if $x_n rightarrow a, n rightarrow infty$ then ${x_n}$ is a Cauchy sequence.




I believe I have found the proof as follows, wondering if there are any simpler methods or added intuition. For me, it makes sense that if a sequence has a limit, then distances between elements in the sequence must be getting smaller, in order for it to converge.



Given $epsilon > 0, exists N_1 s.t. forall n geq N_1:$



$|x_n - a| < frac{epsilon}{2} < epsilon$



and for $m > n geq N_1$ we also have:



$|x_m -a| < frac{epsilon}{2} < epsilon$



Let $N geq N_1$, then $forall n,m geq N$ we have:



$|x_n-x_m| = |x_n - a -x_m+a| < |x_n - a| + |-(x_m -a)| < frac{epsilon}{2} + frac{epsilon}{2} = epsilon$



Therefore ${x_n}$ is a Cauchy sequence.



Also, if a sequence is Cauchy does it always converge? In other words, is it sufficient to check if a sequence is Cauchy to check for convergence.






real-analysis limits proof-verification cauchy-sequences







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    "if a sequence is not Cauchy can it converge?" you literally just proved if it converges it is Cauchy....
    $endgroup$
    – mathworker21
    Jan 29 '17 at 23:17






  • 4




    $begingroup$
    Very good proof. Indeed, if a sequence is convergent, then it is Cauchy (it can't be not Cauchy, you have just proved that!). However, the converse is not true: A space where all Cauchy sequences are convergent, is called a complete space. In complete spaces, Cauchy property is equivalent to convergence. However (for example Riemann integrable functions on $mathbb R$, say) incomplete spaces certainly do exist, but there is an abstract notion of "completion" of a space, which is used time and again to unify these concepts.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Jan 29 '17 at 23:18












  • $begingroup$
    Yes haha sorry, what I meant to ask was if a sequence is Cauchy does it always converge, which астон вілла олоф мэллбэрг explained nicely!
    $endgroup$
    – student_t
    Jan 29 '17 at 23:24












  • $begingroup$
    It is true that a Cauchy sequence in $Bbb R$ converges to a member of $Bbb R.$ This is a consequence of the definition of $Bbb R.$ A Cauchy sequence in $Bbb Q$ will converge in $Bbb R$ but it may or may not converge to a member of $Bbb Q$. So whether a sequence is said to converge depends on the $space$ (e.g. $Bbb Q$ or $Bbb R$ ) that is under consideration.
    $endgroup$
    – DanielWainfleet
    Dec 17 '17 at 5:53










  • $begingroup$
    the title is opposed to your question
    $endgroup$
    – Guy Fsone
    Jan 16 '18 at 12:51














4












4








4


2



$begingroup$



Prove that if $x_n rightarrow a, n rightarrow infty$ then ${x_n}$ is a Cauchy sequence.




I believe I have found the proof as follows, wondering if there are any simpler methods or added intuition. For me, it makes sense that if a sequence has a limit, then distances between elements in the sequence must be getting smaller, in order for it to converge.



Given $epsilon > 0, exists N_1 s.t. forall n geq N_1:$



$|x_n - a| < frac{epsilon}{2} < epsilon$



and for $m > n geq N_1$ we also have:



$|x_m -a| < frac{epsilon}{2} < epsilon$



Let $N geq N_1$, then $forall n,m geq N$ we have:



$|x_n-x_m| = |x_n - a -x_m+a| < |x_n - a| + |-(x_m -a)| < frac{epsilon}{2} + frac{epsilon}{2} = epsilon$



Therefore ${x_n}$ is a Cauchy sequence.



Also, if a sequence is Cauchy does it always converge? In other words, is it sufficient to check if a sequence is Cauchy to check for convergence.






real-analysis limits proof-verification cauchy-sequences







share|cite|improve this question











$endgroup$





Prove that if $x_n rightarrow a, n rightarrow infty$ then ${x_n}$ is a Cauchy sequence.




I believe I have found the proof as follows, wondering if there are any simpler methods or added intuition. For me, it makes sense that if a sequence has a limit, then distances between elements in the sequence must be getting smaller, in order for it to converge.



Given $epsilon > 0, exists N_1 s.t. forall n geq N_1:$



$|x_n - a| < frac{epsilon}{2} < epsilon$



and for $m > n geq N_1$ we also have:



$|x_m -a| < frac{epsilon}{2} < epsilon$



Let $N geq N_1$, then $forall n,m geq N$ we have:



$|x_n-x_m| = |x_n - a -x_m+a| < |x_n - a| + |-(x_m -a)| < frac{epsilon}{2} + frac{epsilon}{2} = epsilon$



Therefore ${x_n}$ is a Cauchy sequence.



Also, if a sequence is Cauchy does it always converge? In other words, is it sufficient to check if a sequence is Cauchy to check for convergence.






real-analysis limits proof-verification cauchy-sequences




real-analysis limits proof-verification cauchy-sequences






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 30 '17 at 17:51







student_t

















asked Jan 29 '17 at 23:15









student_tstudent_t

59339




59339








  • 3




    $begingroup$
    "if a sequence is not Cauchy can it converge?" you literally just proved if it converges it is Cauchy....
    $endgroup$
    – mathworker21
    Jan 29 '17 at 23:17






  • 4




    $begingroup$
    Very good proof. Indeed, if a sequence is convergent, then it is Cauchy (it can't be not Cauchy, you have just proved that!). However, the converse is not true: A space where all Cauchy sequences are convergent, is called a complete space. In complete spaces, Cauchy property is equivalent to convergence. However (for example Riemann integrable functions on $mathbb R$, say) incomplete spaces certainly do exist, but there is an abstract notion of "completion" of a space, which is used time and again to unify these concepts.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Jan 29 '17 at 23:18












  • $begingroup$
    Yes haha sorry, what I meant to ask was if a sequence is Cauchy does it always converge, which астон вілла олоф мэллбэрг explained nicely!
    $endgroup$
    – student_t
    Jan 29 '17 at 23:24












  • $begingroup$
    It is true that a Cauchy sequence in $Bbb R$ converges to a member of $Bbb R.$ This is a consequence of the definition of $Bbb R.$ A Cauchy sequence in $Bbb Q$ will converge in $Bbb R$ but it may or may not converge to a member of $Bbb Q$. So whether a sequence is said to converge depends on the $space$ (e.g. $Bbb Q$ or $Bbb R$ ) that is under consideration.
    $endgroup$
    – DanielWainfleet
    Dec 17 '17 at 5:53










  • $begingroup$
    the title is opposed to your question
    $endgroup$
    – Guy Fsone
    Jan 16 '18 at 12:51














  • 3




    $begingroup$
    "if a sequence is not Cauchy can it converge?" you literally just proved if it converges it is Cauchy....
    $endgroup$
    – mathworker21
    Jan 29 '17 at 23:17






  • 4




    $begingroup$
    Very good proof. Indeed, if a sequence is convergent, then it is Cauchy (it can't be not Cauchy, you have just proved that!). However, the converse is not true: A space where all Cauchy sequences are convergent, is called a complete space. In complete spaces, Cauchy property is equivalent to convergence. However (for example Riemann integrable functions on $mathbb R$, say) incomplete spaces certainly do exist, but there is an abstract notion of "completion" of a space, which is used time and again to unify these concepts.
    $endgroup$
    – астон вілла олоф мэллбэрг
    Jan 29 '17 at 23:18












  • $begingroup$
    Yes haha sorry, what I meant to ask was if a sequence is Cauchy does it always converge, which астон вілла олоф мэллбэрг explained nicely!
    $endgroup$
    – student_t
    Jan 29 '17 at 23:24












  • $begingroup$
    It is true that a Cauchy sequence in $Bbb R$ converges to a member of $Bbb R.$ This is a consequence of the definition of $Bbb R.$ A Cauchy sequence in $Bbb Q$ will converge in $Bbb R$ but it may or may not converge to a member of $Bbb Q$. So whether a sequence is said to converge depends on the $space$ (e.g. $Bbb Q$ or $Bbb R$ ) that is under consideration.
    $endgroup$
    – DanielWainfleet
    Dec 17 '17 at 5:53










  • $begingroup$
    the title is opposed to your question
    $endgroup$
    – Guy Fsone
    Jan 16 '18 at 12:51








3




3




$begingroup$
"if a sequence is not Cauchy can it converge?" you literally just proved if it converges it is Cauchy....
$endgroup$
– mathworker21
Jan 29 '17 at 23:17




$begingroup$
"if a sequence is not Cauchy can it converge?" you literally just proved if it converges it is Cauchy....
$endgroup$
– mathworker21
Jan 29 '17 at 23:17




4




4




$begingroup$
Very good proof. Indeed, if a sequence is convergent, then it is Cauchy (it can't be not Cauchy, you have just proved that!). However, the converse is not true: A space where all Cauchy sequences are convergent, is called a complete space. In complete spaces, Cauchy property is equivalent to convergence. However (for example Riemann integrable functions on $mathbb R$, say) incomplete spaces certainly do exist, but there is an abstract notion of "completion" of a space, which is used time and again to unify these concepts.
$endgroup$
– астон вілла олоф мэллбэрг
Jan 29 '17 at 23:18






$begingroup$
Very good proof. Indeed, if a sequence is convergent, then it is Cauchy (it can't be not Cauchy, you have just proved that!). However, the converse is not true: A space where all Cauchy sequences are convergent, is called a complete space. In complete spaces, Cauchy property is equivalent to convergence. However (for example Riemann integrable functions on $mathbb R$, say) incomplete spaces certainly do exist, but there is an abstract notion of "completion" of a space, which is used time and again to unify these concepts.
$endgroup$
– астон вілла олоф мэллбэрг
Jan 29 '17 at 23:18














$begingroup$
Yes haha sorry, what I meant to ask was if a sequence is Cauchy does it always converge, which астон вілла олоф мэллбэрг explained nicely!
$endgroup$
– student_t
Jan 29 '17 at 23:24






$begingroup$
Yes haha sorry, what I meant to ask was if a sequence is Cauchy does it always converge, which астон вілла олоф мэллбэрг explained nicely!
$endgroup$
– student_t
Jan 29 '17 at 23:24














$begingroup$
It is true that a Cauchy sequence in $Bbb R$ converges to a member of $Bbb R.$ This is a consequence of the definition of $Bbb R.$ A Cauchy sequence in $Bbb Q$ will converge in $Bbb R$ but it may or may not converge to a member of $Bbb Q$. So whether a sequence is said to converge depends on the $space$ (e.g. $Bbb Q$ or $Bbb R$ ) that is under consideration.
$endgroup$
– DanielWainfleet
Dec 17 '17 at 5:53




$begingroup$
It is true that a Cauchy sequence in $Bbb R$ converges to a member of $Bbb R.$ This is a consequence of the definition of $Bbb R.$ A Cauchy sequence in $Bbb Q$ will converge in $Bbb R$ but it may or may not converge to a member of $Bbb Q$. So whether a sequence is said to converge depends on the $space$ (e.g. $Bbb Q$ or $Bbb R$ ) that is under consideration.
$endgroup$
– DanielWainfleet
Dec 17 '17 at 5:53












$begingroup$
the title is opposed to your question
$endgroup$
– Guy Fsone
Jan 16 '18 at 12:51




$begingroup$
the title is opposed to your question
$endgroup$
– Guy Fsone
Jan 16 '18 at 12:51










1 Answer
1






active

oldest

votes


















0












$begingroup$

With response to the comments above, it is not always true that a Cauchy sequence is convergent. For example, if we think of metric spaces,
yes it is true that any Cauchy sequence in $(R, mathrm{d}E)$ is convergent and any Cauchy sequence in $(X, mathrm{d}delta)$ is eventually constant, so convergent. We have the counter example however: Cauchy does not imply convergent in $(0, infty)$, $d(x, y) = |x − y|$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I guess I should have clarified. This is for metric spaces.
    $endgroup$
    – student_t
    Dec 16 '17 at 23:00






  • 1




    $begingroup$
    It's not true for all metric spaces. $mathbb Q$ with the normal metric is a metric space where it is not true.
    $endgroup$
    – fleablood
    Feb 20 '18 at 17:04










  • $begingroup$
    Or try $Bbb R-{0}$.
    $endgroup$
    – Ted Shifrin
    Nov 7 '18 at 18:06











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1 Answer
1






active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0












$begingroup$

With response to the comments above, it is not always true that a Cauchy sequence is convergent. For example, if we think of metric spaces,
yes it is true that any Cauchy sequence in $(R, mathrm{d}E)$ is convergent and any Cauchy sequence in $(X, mathrm{d}delta)$ is eventually constant, so convergent. We have the counter example however: Cauchy does not imply convergent in $(0, infty)$, $d(x, y) = |x − y|$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I guess I should have clarified. This is for metric spaces.
    $endgroup$
    – student_t
    Dec 16 '17 at 23:00






  • 1




    $begingroup$
    It's not true for all metric spaces. $mathbb Q$ with the normal metric is a metric space where it is not true.
    $endgroup$
    – fleablood
    Feb 20 '18 at 17:04










  • $begingroup$
    Or try $Bbb R-{0}$.
    $endgroup$
    – Ted Shifrin
    Nov 7 '18 at 18:06
















0












$begingroup$

With response to the comments above, it is not always true that a Cauchy sequence is convergent. For example, if we think of metric spaces,
yes it is true that any Cauchy sequence in $(R, mathrm{d}E)$ is convergent and any Cauchy sequence in $(X, mathrm{d}delta)$ is eventually constant, so convergent. We have the counter example however: Cauchy does not imply convergent in $(0, infty)$, $d(x, y) = |x − y|$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I guess I should have clarified. This is for metric spaces.
    $endgroup$
    – student_t
    Dec 16 '17 at 23:00






  • 1




    $begingroup$
    It's not true for all metric spaces. $mathbb Q$ with the normal metric is a metric space where it is not true.
    $endgroup$
    – fleablood
    Feb 20 '18 at 17:04










  • $begingroup$
    Or try $Bbb R-{0}$.
    $endgroup$
    – Ted Shifrin
    Nov 7 '18 at 18:06














0












0








0





$begingroup$

With response to the comments above, it is not always true that a Cauchy sequence is convergent. For example, if we think of metric spaces,
yes it is true that any Cauchy sequence in $(R, mathrm{d}E)$ is convergent and any Cauchy sequence in $(X, mathrm{d}delta)$ is eventually constant, so convergent. We have the counter example however: Cauchy does not imply convergent in $(0, infty)$, $d(x, y) = |x − y|$.






share|cite|improve this answer











$endgroup$



With response to the comments above, it is not always true that a Cauchy sequence is convergent. For example, if we think of metric spaces,
yes it is true that any Cauchy sequence in $(R, mathrm{d}E)$ is convergent and any Cauchy sequence in $(X, mathrm{d}delta)$ is eventually constant, so convergent. We have the counter example however: Cauchy does not imply convergent in $(0, infty)$, $d(x, y) = |x − y|$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 16 '17 at 23:25









Xander Henderson

14.8k103555




14.8k103555










answered Dec 16 '17 at 22:59









WronskianaWronskiana

1




1












  • $begingroup$
    I guess I should have clarified. This is for metric spaces.
    $endgroup$
    – student_t
    Dec 16 '17 at 23:00






  • 1




    $begingroup$
    It's not true for all metric spaces. $mathbb Q$ with the normal metric is a metric space where it is not true.
    $endgroup$
    – fleablood
    Feb 20 '18 at 17:04










  • $begingroup$
    Or try $Bbb R-{0}$.
    $endgroup$
    – Ted Shifrin
    Nov 7 '18 at 18:06


















  • $begingroup$
    I guess I should have clarified. This is for metric spaces.
    $endgroup$
    – student_t
    Dec 16 '17 at 23:00






  • 1




    $begingroup$
    It's not true for all metric spaces. $mathbb Q$ with the normal metric is a metric space where it is not true.
    $endgroup$
    – fleablood
    Feb 20 '18 at 17:04










  • $begingroup$
    Or try $Bbb R-{0}$.
    $endgroup$
    – Ted Shifrin
    Nov 7 '18 at 18:06
















$begingroup$
I guess I should have clarified. This is for metric spaces.
$endgroup$
– student_t
Dec 16 '17 at 23:00




$begingroup$
I guess I should have clarified. This is for metric spaces.
$endgroup$
– student_t
Dec 16 '17 at 23:00




1




1




$begingroup$
It's not true for all metric spaces. $mathbb Q$ with the normal metric is a metric space where it is not true.
$endgroup$
– fleablood
Feb 20 '18 at 17:04




$begingroup$
It's not true for all metric spaces. $mathbb Q$ with the normal metric is a metric space where it is not true.
$endgroup$
– fleablood
Feb 20 '18 at 17:04












$begingroup$
Or try $Bbb R-{0}$.
$endgroup$
– Ted Shifrin
Nov 7 '18 at 18:06




$begingroup$
Or try $Bbb R-{0}$.
$endgroup$
– Ted Shifrin
Nov 7 '18 at 18:06


















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Musa

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Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
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