Existence of natural number in the set $left textrm Q leq sin(x) < 1right $, $0 < Q < 1$

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Suppose $Q in (0,1)$. Then clearly, $exists$ $x in mathbbR$, $ni$ $ sin(x) = Q$. We define a set



beginequation*
P = left textrm Q leq sin(x) < 1 right
endequation*



This set will be non-empty. My question is (with Archana Puran Singh accent), is this set $P cap mathbbN$ non-empty ? Or is there some $Q$ for which the set $P cap mathbbN$ is empty ? If we take $Q$ near 1, will it ensure that $P cap mathbbN = varnothing$ ?



My MO is that I need to find some $x,y in S$, $0<x<y$ and $y-x > 1$. If I can prove this, then $P cap mathbbN$ being non-empty will easily follow.




real-analysis sequences-and-series




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  • If the set P contains only those real numbers between Q and 1, where Q > 0, then how can there be any natural numbers in P?
    – esotechnica
    Aug 19 at 9:04











  • I don't understand the final paragraph: what is the set $S$?
    – Barry Cipra
    Aug 19 at 9:45










  • On a lighter note: What does (with Archana Puran Singh accent) mean? A joke?
    – user539887
    Aug 19 at 9:47










  • @esotechnica The OP asks about the set of reals mapped onto $[Q,1]$ by the sine function.
    – user539887
    Aug 19 at 9:50















up vote
0
down vote

favorite












Suppose $Q in (0,1)$. Then clearly, $exists$ $x in mathbbR$, $ni$ $ sin(x) = Q$. We define a set



beginequation*
P = left textrm Q leq sin(x) < 1 right
endequation*



This set will be non-empty. My question is (with Archana Puran Singh accent), is this set $P cap mathbbN$ non-empty ? Or is there some $Q$ for which the set $P cap mathbbN$ is empty ? If we take $Q$ near 1, will it ensure that $P cap mathbbN = varnothing$ ?



My MO is that I need to find some $x,y in S$, $0<x<y$ and $y-x > 1$. If I can prove this, then $P cap mathbbN$ being non-empty will easily follow.




real-analysis sequences-and-series




share|cite|improve this question






















  • If the set P contains only those real numbers between Q and 1, where Q > 0, then how can there be any natural numbers in P?
    – esotechnica
    Aug 19 at 9:04











  • I don't understand the final paragraph: what is the set $S$?
    – Barry Cipra
    Aug 19 at 9:45










  • On a lighter note: What does (with Archana Puran Singh accent) mean? A joke?
    – user539887
    Aug 19 at 9:47










  • @esotechnica The OP asks about the set of reals mapped onto $[Q,1]$ by the sine function.
    – user539887
    Aug 19 at 9:50













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Suppose $Q in (0,1)$. Then clearly, $exists$ $x in mathbbR$, $ni$ $ sin(x) = Q$. We define a set



beginequation*
P = left textrm Q leq sin(x) < 1 right
endequation*



This set will be non-empty. My question is (with Archana Puran Singh accent), is this set $P cap mathbbN$ non-empty ? Or is there some $Q$ for which the set $P cap mathbbN$ is empty ? If we take $Q$ near 1, will it ensure that $P cap mathbbN = varnothing$ ?



My MO is that I need to find some $x,y in S$, $0<x<y$ and $y-x > 1$. If I can prove this, then $P cap mathbbN$ being non-empty will easily follow.




real-analysis sequences-and-series




share|cite|improve this question














Suppose $Q in (0,1)$. Then clearly, $exists$ $x in mathbbR$, $ni$ $ sin(x) = Q$. We define a set



beginequation*
P = left textrm Q leq sin(x) < 1 right
endequation*



This set will be non-empty. My question is (with Archana Puran Singh accent), is this set $P cap mathbbN$ non-empty ? Or is there some $Q$ for which the set $P cap mathbbN$ is empty ? If we take $Q$ near 1, will it ensure that $P cap mathbbN = varnothing$ ?



My MO is that I need to find some $x,y in S$, $0<x<y$ and $y-x > 1$. If I can prove this, then $P cap mathbbN$ being non-empty will easily follow.





real-analysis sequences-and-series





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edited Aug 19 at 9:36









rtybase

8,98221433




8,98221433










asked Aug 19 at 8:21









Minto P

174




174











  • If the set P contains only those real numbers between Q and 1, where Q > 0, then how can there be any natural numbers in P?
    – esotechnica
    Aug 19 at 9:04











  • I don't understand the final paragraph: what is the set $S$?
    – Barry Cipra
    Aug 19 at 9:45










  • On a lighter note: What does (with Archana Puran Singh accent) mean? A joke?
    – user539887
    Aug 19 at 9:47










  • @esotechnica The OP asks about the set of reals mapped onto $[Q,1]$ by the sine function.
    – user539887
    Aug 19 at 9:50

















  • If the set P contains only those real numbers between Q and 1, where Q > 0, then how can there be any natural numbers in P?
    – esotechnica
    Aug 19 at 9:04











  • I don't understand the final paragraph: what is the set $S$?
    – Barry Cipra
    Aug 19 at 9:45










  • On a lighter note: What does (with Archana Puran Singh accent) mean? A joke?
    – user539887
    Aug 19 at 9:47










  • @esotechnica The OP asks about the set of reals mapped onto $[Q,1]$ by the sine function.
    – user539887
    Aug 19 at 9:50
















If the set P contains only those real numbers between Q and 1, where Q > 0, then how can there be any natural numbers in P?
– esotechnica
Aug 19 at 9:04





If the set P contains only those real numbers between Q and 1, where Q > 0, then how can there be any natural numbers in P?
– esotechnica
Aug 19 at 9:04













I don't understand the final paragraph: what is the set $S$?
– Barry Cipra
Aug 19 at 9:45




I don't understand the final paragraph: what is the set $S$?
– Barry Cipra
Aug 19 at 9:45












On a lighter note: What does (with Archana Puran Singh accent) mean? A joke?
– user539887
Aug 19 at 9:47




On a lighter note: What does (with Archana Puran Singh accent) mean? A joke?
– user539887
Aug 19 at 9:47












@esotechnica The OP asks about the set of reals mapped onto $[Q,1]$ by the sine function.
– user539887
Aug 19 at 9:50





@esotechnica The OP asks about the set of reals mapped onto $[Q,1]$ by the sine function.
– user539887
Aug 19 at 9:50











1 Answer
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Use the fact that




$sinn mid n in mathbbN$ is dense in $[-1,1]$




proved here. As a result $forall x,y in mathbbR$ with $Q<x<y<1$, $exists n in mathbbN$ such that $Q<x<sinn<y<1$, which means $n in P$ and $Pcap mathbbNne varnothing$.






share|cite|improve this answer




















  • But is there any way to prove this problem without using the denseness property of $left textrm n in mathbbN right$ ?
    – Minto P
    Aug 20 at 7:16










  • It will be a tedious proof of course, but you can look at continued fractions and convergents of $fracpi2$.
    – rtybase
    Aug 20 at 8:36











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1 Answer
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1 Answer
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active

oldest

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active

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up vote
0
down vote













Use the fact that




$sinn mid n in mathbbN$ is dense in $[-1,1]$




proved here. As a result $forall x,y in mathbbR$ with $Q<x<y<1$, $exists n in mathbbN$ such that $Q<x<sinn<y<1$, which means $n in P$ and $Pcap mathbbNne varnothing$.






share|cite|improve this answer




















  • But is there any way to prove this problem without using the denseness property of $left textrm n in mathbbN right$ ?
    – Minto P
    Aug 20 at 7:16










  • It will be a tedious proof of course, but you can look at continued fractions and convergents of $fracpi2$.
    – rtybase
    Aug 20 at 8:36















up vote
0
down vote













Use the fact that




$sinn mid n in mathbbN$ is dense in $[-1,1]$




proved here. As a result $forall x,y in mathbbR$ with $Q<x<y<1$, $exists n in mathbbN$ such that $Q<x<sinn<y<1$, which means $n in P$ and $Pcap mathbbNne varnothing$.






share|cite|improve this answer




















  • But is there any way to prove this problem without using the denseness property of $left textrm n in mathbbN right$ ?
    – Minto P
    Aug 20 at 7:16










  • It will be a tedious proof of course, but you can look at continued fractions and convergents of $fracpi2$.
    – rtybase
    Aug 20 at 8:36













up vote
0
down vote










up vote
0
down vote









Use the fact that




$sinn mid n in mathbbN$ is dense in $[-1,1]$




proved here. As a result $forall x,y in mathbbR$ with $Q<x<y<1$, $exists n in mathbbN$ such that $Q<x<sinn<y<1$, which means $n in P$ and $Pcap mathbbNne varnothing$.






share|cite|improve this answer












Use the fact that




$sinn mid n in mathbbN$ is dense in $[-1,1]$




proved here. As a result $forall x,y in mathbbR$ with $Q<x<y<1$, $exists n in mathbbN$ such that $Q<x<sinn<y<1$, which means $n in P$ and $Pcap mathbbNne varnothing$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Aug 19 at 9:25









rtybase

8,98221433




8,98221433











  • But is there any way to prove this problem without using the denseness property of $left textrm n in mathbbN right$ ?
    – Minto P
    Aug 20 at 7:16










  • It will be a tedious proof of course, but you can look at continued fractions and convergents of $fracpi2$.
    – rtybase
    Aug 20 at 8:36

















  • But is there any way to prove this problem without using the denseness property of $left textrm n in mathbbN right$ ?
    – Minto P
    Aug 20 at 7:16










  • It will be a tedious proof of course, but you can look at continued fractions and convergents of $fracpi2$.
    – rtybase
    Aug 20 at 8:36
















But is there any way to prove this problem without using the denseness property of $left textrm n in mathbbN right$ ?
– Minto P
Aug 20 at 7:16




But is there any way to prove this problem without using the denseness property of $left textrm n in mathbbN right$ ?
– Minto P
Aug 20 at 7:16












It will be a tedious proof of course, but you can look at continued fractions and convergents of $fracpi2$.
– rtybase
Aug 20 at 8:36





It will be a tedious proof of course, but you can look at continued fractions and convergents of $fracpi2$.
– rtybase
Aug 20 at 8:36













 

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