Convergent Sequences problem

  • Thread starter pbxed
  • Start date
  • #1
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Homework Statement



Let f: N -> N be a bijective map. for n Є N

a sub n = 1 / f(n)

Show that the sequence (a sub n) converges to zero.


Homework Equations





The Attempt at a Solution



Basically I have been stuck on this problem for hours now and have read through my notes and the course textbook numerous times and am still not getting anywhere. I sort of get how to apply the definition of a convergent sequences to a few other questions but my understanding of this topic is clearly pretty poor.

Can someone give me a hint either as a question to get me started or some material available that would clarify this for me ?

Thanks in advance

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
22,129
3,298
So you'll want to show that

[tex] \forall \epsilon >0~\exists n_0~\forall n>n_0:~|1/f(n)|<\epsilon [/tex]

This is equivalent with (take [tex]N=1/\epsilon [/tex]):

[tex]\forall N>0~\exists n_0~\forall n>n_0:~|f(n)|>N[/tex]

So you'll have to show that, from a certain moment on, the sequence gets bigger then N (and this forall N). But since f is a bijection, it is true that, from a certain [tex]n_0[/tex], f has already assumed all elements smaller then N. So from that moment on, f is always larger then N. Which we needed to prove.

Ï hope you understand my ramblings :smile:
 
  • #3
14
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is it rude of me to say that I don't really understand your reply and ask for further clarification, either from yourself on someone else?
 

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