What is a sequence that gets closer and closer to a without converging to a?

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Find a sequence $a_n$ and a real number $a$ so that $|a_n+1-a| < |a_n-a|$ for each n, but $a_n$ does not converge to $a$. So the sequence gets closer and closer to $a$ without converging to $a$.



I have seen many questions posted here asking how to find a sequence that converges to $a$ without getting closer and closer to $a$, so this question is kind of the opposite. My intuition is telling me that I need to find a function that increases slower and slower but continues to increase. Any and all help is appreciated- thank you!






real-analysis sequences-and-series analysis convergence







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  • 1




    Hint: it can converge to something other than a. Say it converges to 1. Then a sequence that gets closer and closer but always more than 1 gets closer and closer to any number less than 1. Btw, I've never seen a question of a sequence that converges but doesn't get close.
    – fleablood
    Sep 11 at 2:44















up vote
0
down vote

favorite












Find a sequence $a_n$ and a real number $a$ so that $|a_n+1-a| < |a_n-a|$ for each n, but $a_n$ does not converge to $a$. So the sequence gets closer and closer to $a$ without converging to $a$.



I have seen many questions posted here asking how to find a sequence that converges to $a$ without getting closer and closer to $a$, so this question is kind of the opposite. My intuition is telling me that I need to find a function that increases slower and slower but continues to increase. Any and all help is appreciated- thank you!






real-analysis sequences-and-series analysis convergence







share|cite|improve this question

















  • 1




    Hint: it can converge to something other than a. Say it converges to 1. Then a sequence that gets closer and closer but always more than 1 gets closer and closer to any number less than 1. Btw, I've never seen a question of a sequence that converges but doesn't get close.
    – fleablood
    Sep 11 at 2:44













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Find a sequence $a_n$ and a real number $a$ so that $|a_n+1-a| < |a_n-a|$ for each n, but $a_n$ does not converge to $a$. So the sequence gets closer and closer to $a$ without converging to $a$.



I have seen many questions posted here asking how to find a sequence that converges to $a$ without getting closer and closer to $a$, so this question is kind of the opposite. My intuition is telling me that I need to find a function that increases slower and slower but continues to increase. Any and all help is appreciated- thank you!






real-analysis sequences-and-series analysis convergence







share|cite|improve this question













Find a sequence $a_n$ and a real number $a$ so that $|a_n+1-a| < |a_n-a|$ for each n, but $a_n$ does not converge to $a$. So the sequence gets closer and closer to $a$ without converging to $a$.



I have seen many questions posted here asking how to find a sequence that converges to $a$ without getting closer and closer to $a$, so this question is kind of the opposite. My intuition is telling me that I need to find a function that increases slower and slower but continues to increase. Any and all help is appreciated- thank you!






real-analysis sequences-and-series analysis convergence




real-analysis sequences-and-series analysis convergence






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asked Sep 11 at 2:20









Michael

347




347







  • 1




    Hint: it can converge to something other than a. Say it converges to 1. Then a sequence that gets closer and closer but always more than 1 gets closer and closer to any number less than 1. Btw, I've never seen a question of a sequence that converges but doesn't get close.
    – fleablood
    Sep 11 at 2:44













  • 1




    Hint: it can converge to something other than a. Say it converges to 1. Then a sequence that gets closer and closer but always more than 1 gets closer and closer to any number less than 1. Btw, I've never seen a question of a sequence that converges but doesn't get close.
    – fleablood
    Sep 11 at 2:44








1




1




Hint: it can converge to something other than a. Say it converges to 1. Then a sequence that gets closer and closer but always more than 1 gets closer and closer to any number less than 1. Btw, I've never seen a question of a sequence that converges but doesn't get close.
– fleablood
Sep 11 at 2:44





Hint: it can converge to something other than a. Say it converges to 1. Then a sequence that gets closer and closer but always more than 1 gets closer and closer to any number less than 1. Btw, I've never seen a question of a sequence that converges but doesn't get close.
– fleablood
Sep 11 at 2:44











4 Answers
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2
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You can simply take $a=0$ and $a_n=1+frac1n$.






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  • Thank you for the clear example!
    – Michael
    Sep 11 at 3:39

















up vote
2
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Hint: If a sequence converges to $1$ from above, the distances between each of its terms and $0$ decrease.






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  • Thank you. Now this question seems so simple.
    – Michael
    Sep 11 at 3:39

















up vote
2
down vote













Your intuition appears to be leading you in a good direction. An increasing sequence is a good thing to try.



Note that an increasing sequence that converges to a number has a property you're looking for: it increases slower and slower.



Try to construct an actual increasing, converging sequence $a_n.$ Then consider some of the numbers that $a_n$ does not converge to. Could any of them be the number $a$ you need?






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  • Thank you for the guidance. I appreciate answers that do not immediately reveal an answer!
    – Michael
    Sep 11 at 3:39

















up vote
1
down vote













For example,
$$a_n=frac1nquadhboxandquad a=-2018 .$$






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  • Thank you for the example!
    – Michael
    Sep 11 at 3:39










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4 Answers
4






active

oldest

votes








4 Answers
4






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
2
down vote













You can simply take $a=0$ and $a_n=1+frac1n$.






share|cite|improve this answer




















  • Thank you for the clear example!
    – Michael
    Sep 11 at 3:39














up vote
2
down vote













You can simply take $a=0$ and $a_n=1+frac1n$.






share|cite|improve this answer




















  • Thank you for the clear example!
    – Michael
    Sep 11 at 3:39












up vote
2
down vote










up vote
2
down vote









You can simply take $a=0$ and $a_n=1+frac1n$.






share|cite|improve this answer












You can simply take $a=0$ and $a_n=1+frac1n$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Sep 11 at 2:23









Eclipse Sun

6,7021337




6,7021337











  • Thank you for the clear example!
    – Michael
    Sep 11 at 3:39
















  • Thank you for the clear example!
    – Michael
    Sep 11 at 3:39















Thank you for the clear example!
– Michael
Sep 11 at 3:39




Thank you for the clear example!
– Michael
Sep 11 at 3:39










up vote
2
down vote













Hint: If a sequence converges to $1$ from above, the distances between each of its terms and $0$ decrease.






share|cite|improve this answer




















  • Thank you. Now this question seems so simple.
    – Michael
    Sep 11 at 3:39














up vote
2
down vote













Hint: If a sequence converges to $1$ from above, the distances between each of its terms and $0$ decrease.






share|cite|improve this answer




















  • Thank you. Now this question seems so simple.
    – Michael
    Sep 11 at 3:39












up vote
2
down vote










up vote
2
down vote









Hint: If a sequence converges to $1$ from above, the distances between each of its terms and $0$ decrease.






share|cite|improve this answer












Hint: If a sequence converges to $1$ from above, the distances between each of its terms and $0$ decrease.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Sep 11 at 2:23









Carl Schildkraut

10.4k11438




10.4k11438











  • Thank you. Now this question seems so simple.
    – Michael
    Sep 11 at 3:39
















  • Thank you. Now this question seems so simple.
    – Michael
    Sep 11 at 3:39















Thank you. Now this question seems so simple.
– Michael
Sep 11 at 3:39




Thank you. Now this question seems so simple.
– Michael
Sep 11 at 3:39










up vote
2
down vote













Your intuition appears to be leading you in a good direction. An increasing sequence is a good thing to try.



Note that an increasing sequence that converges to a number has a property you're looking for: it increases slower and slower.



Try to construct an actual increasing, converging sequence $a_n.$ Then consider some of the numbers that $a_n$ does not converge to. Could any of them be the number $a$ you need?






share|cite|improve this answer




















  • Thank you for the guidance. I appreciate answers that do not immediately reveal an answer!
    – Michael
    Sep 11 at 3:39














up vote
2
down vote













Your intuition appears to be leading you in a good direction. An increasing sequence is a good thing to try.



Note that an increasing sequence that converges to a number has a property you're looking for: it increases slower and slower.



Try to construct an actual increasing, converging sequence $a_n.$ Then consider some of the numbers that $a_n$ does not converge to. Could any of them be the number $a$ you need?






share|cite|improve this answer




















  • Thank you for the guidance. I appreciate answers that do not immediately reveal an answer!
    – Michael
    Sep 11 at 3:39












up vote
2
down vote










up vote
2
down vote









Your intuition appears to be leading you in a good direction. An increasing sequence is a good thing to try.



Note that an increasing sequence that converges to a number has a property you're looking for: it increases slower and slower.



Try to construct an actual increasing, converging sequence $a_n.$ Then consider some of the numbers that $a_n$ does not converge to. Could any of them be the number $a$ you need?






share|cite|improve this answer












Your intuition appears to be leading you in a good direction. An increasing sequence is a good thing to try.



Note that an increasing sequence that converges to a number has a property you're looking for: it increases slower and slower.



Try to construct an actual increasing, converging sequence $a_n.$ Then consider some of the numbers that $a_n$ does not converge to. Could any of them be the number $a$ you need?







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Sep 11 at 2:37









David K

50.8k340113




50.8k340113











  • Thank you for the guidance. I appreciate answers that do not immediately reveal an answer!
    – Michael
    Sep 11 at 3:39
















  • Thank you for the guidance. I appreciate answers that do not immediately reveal an answer!
    – Michael
    Sep 11 at 3:39















Thank you for the guidance. I appreciate answers that do not immediately reveal an answer!
– Michael
Sep 11 at 3:39




Thank you for the guidance. I appreciate answers that do not immediately reveal an answer!
– Michael
Sep 11 at 3:39










up vote
1
down vote













For example,
$$a_n=frac1nquadhboxandquad a=-2018 .$$






share|cite|improve this answer




















  • Thank you for the example!
    – Michael
    Sep 11 at 3:39














up vote
1
down vote













For example,
$$a_n=frac1nquadhboxandquad a=-2018 .$$






share|cite|improve this answer




















  • Thank you for the example!
    – Michael
    Sep 11 at 3:39












up vote
1
down vote










up vote
1
down vote









For example,
$$a_n=frac1nquadhboxandquad a=-2018 .$$






share|cite|improve this answer












For example,
$$a_n=frac1nquadhboxandquad a=-2018 .$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Sep 11 at 2:23









David

67.2k663126




67.2k663126











  • Thank you for the example!
    – Michael
    Sep 11 at 3:39
















  • Thank you for the example!
    – Michael
    Sep 11 at 3:39















Thank you for the example!
– Michael
Sep 11 at 3:39




Thank you for the example!
– Michael
Sep 11 at 3:39

















 

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