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

[kingstrick]

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 \leq 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 - \frac{1}{d} 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.

 

[pasmith]

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.

 

[kingstrick]

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

 

[Ray Vickson]

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).

 

[pasmith]

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 a &gt; 0 there exists a positive integer n such that n^{-1} &lt; a, or do you have to prove that as well?

 

[kingstrick]

yes

Are you allowed to assume that for every a &gt; 0 there exists a positive integer n such that n^{-1} &lt; a, or do you have to prove that as well?

yes. This was one of our axioms