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Limit point of Sn := {1-1/n} is 1

  • Thread starter kingstrick
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  • #1
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Homework Statement




Show that the limit point of Sn:={1-1/n} is 1.

Homework Equations



We are prohibited from using epsilon and delta

The Attempt at a Solution



Let Sn:= {1-1/n} and U be any open interval from (a,b) where a<1<b. Observe that Sn is always [itex]\leq[/itex] 1. Since a<1 is linearly ordered, there is a positive number d between a and 1 such that a<d<1 and 1/d > 1. Then since d < 1,
d-1 < 1-1
d-1 < 0
(d-1)/d < 0 --> (d-1)/d = 1 - [itex]\frac{1}{d}[/itex] and ...

i am stuck, it appears that i shown my Sn is always less then zero making my limit point zero not one. Any help would be appreciated.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
pasmith
Homework Helper
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Homework Statement




Show that the limit point of Sn:={1-1/n} is 1.

Homework Equations



We are prohibited from using epsilon and delta
This appears to be an invitation to prove the result from basic properties of convergent sequences and already known limits.
 
  • #3
108
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Here are the definitions that we must use. We are prohibited also from using monotonic and bounded.


definition:

The statement that the point sequence p1, p2, . . . converges to the point x means that if S is an open interval containing x then there is a positive integer N such that if n is a positive integer and n ≥ N then pn ∈ S.

and

The statement that the sequence p1, p2, p3, . . . converges means that there is a point x such that p1, p2, p3, . . . converges
 
  • #4
Ray Vickson
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Here are the definitions that we must use. We are prohibited also from using monotonic and bounded.


definition:

The statement that the point sequence p1, p2, . . . converges to the point x means that if S is an open interval containing x then there is a positive integer N such that if n is a positive integer and n ≥ N then pn ∈ S.

and

The statement that the sequence p1, p2, p3, . . . converges means that there is a point x such that p1, p2, p3, . . . converges
So, basically, you are using ε and N (not δ, of course, since n is not staying finite).
 
  • #5
pasmith
Homework Helper
1,738
410
Here are the definitions that we must use. We are prohibited also from using monotonic and bounded.


definition:

The statement that the point sequence p1, p2, . . . converges to the point x means that if S is an open interval containing x then there is a positive integer N such that if n is a positive integer and n ≥ N then pn ∈ S.

Are you allowed to assume that for every [itex]a > 0[/itex] there exists a positive integer [itex]n[/itex] such that [itex]n^{-1} < a[/itex], or do you have to prove that as well?
 
  • #6
108
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yes

Are you allowed to assume that for every [itex]a > 0[/itex] there exists a positive integer [itex]n[/itex] such that [itex]n^{-1} < a[/itex], or do you have to prove that as well?
yes. This was one of our axioms
 

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