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Finding a convergent subsequence does the sequence need to be bounded

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  • #1
ppy
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Homework Statement



2.11. Determine (explicitly) a convergent subsequence of the sequence in R2 given for n =
1; 2; : : : by
xn =(e[itex]^{n}[/itex]sin(n[itex]\pi[/itex]/7),((4n+3/3n+4)cos(n[itex]\pi[/itex]/3))



I know that the Bolzano-weierstrass theorem says that every bounded sequence has a convergent subsequence. I thought I should first check that the xn is bounded by checking each individual co-ordinate. however isn't x[itex]_{n,1}[/itex] not bounded? Therefore surely x[itex]_{n}[/itex] cannot have a convergent subsequence? as doesn't it just go to infinity? Help needed urgently!!

Thanks
 

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  • #2
Dick
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Homework Helper
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Homework Statement



2.11. Determine (explicitly) a convergent subsequence of the sequence in R2 given for n =
1; 2; : : : by
xn =(e[itex]^{n}[/itex]sin(n[itex]\pi[/itex]/7),((4n+3/3n+4)cos(n[itex]\pi[/itex]/3))



I know that the Bolzano-weierstrass theorem says that every bounded sequence has a convergent subsequence. I thought I should first check that the xn is bounded by checking each individual co-ordinate. however isn't x[itex]_{n,1}[/itex] not bounded? Therefore surely x[itex]_{n}[/itex] cannot have a convergent subsequence? as doesn't it just go to infinity? Help needed urgently!!

Thanks
Being unbounded doesn't necessarily mean a sequence has no convergent sequences. Think about the x and y coordinates separately. Can you find a convergent sequence of the x coordinate.
 
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I know that the Bolzano-weierstrass theorem says that every bounded sequence has a convergent subsequence. I thought I should first check that the xn is bounded by checking each individual co-ordinate. however isn't x[itex]_{n,1}[/itex] not bounded? Therefore surely x[itex]_{n}[/itex] cannot have a convergent subsequence?
The B-W theorem doesn't say that if a sequence has a convergent subsequence it is bounded. Look at the first component. We know a priori that ##e^n## is headed for ##\infty##. So the only hope for a convergent subsequence on that component is to pick values of n where the sin pulls things down. What do you know about sin(n##\pi##)?
 

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