# 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.

## 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.

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

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).

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 $a > 0$ there exists a positive integer $n$ such that $n^{-1} < a$, or do you have to prove that as well?

yes

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

yes. This was one of our axioms