Limit Points

Definition (Limit Points).

Let be a sequence of real number. Then is said to be a limit point if there exist a subgroup of sequence of converging to

Examples

has no limit points
has exactly one limit points
has exactly two limit points
let and we define will have infinitely many limit point

Claim.

The only limit points of The above example The above example is where

Proof.
Zero is the limit of as . And is the limit of as . Thus are all limit points. We will show these are the only limit points.

Let . Then there is an such that the interval has no point of the form (or 0).

Therefore, the sum , with least one of or less than , must be at a distance of at least from . So the only such numbers in the interval must have and . So and . Hence there are only a finite number of numbers of the form in the interval . Thus is not a limit point.
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#25-sep
If a sequence is bounded than the set of limit points of the sequence is also bounded

Definition (limit Superior and Limit Inferior).

For a bounded sequence the is called as the limit Inferior
For a bounded sequence the is called as the superior Inferior

Notation

Theorem.

Let be a real sequence then $ε$

ε ε ε ε

Proof.
fix $ε$ $ε$ such that $ε$ $ε$

ε ε ε ε ε

Conversely
consider

ε

such that

ε ε ε

Inductively we can find infinite values of

if conversely such that $ε$ is finite
fix $ε$ $ε$ is infinite for $ε$
$ε$ is finite for $ε$ $ε$
$ε$
$ε$
$ε$

Claim (

ε ε ε

this shows

Theorem.

Let be a real sequence then $ε$

ε ε ε ε

conversely if is finite and is infinite for some and all $ε$ then is the liminf

Theorem.

Let be a bounded sequence of Then Limsup of is a limit point of and if is limt point of then

Theorem.

Let be a bounded real sequence of then all the limit points of lies in

Theorem.

be a bounded set of real numbers then the set of limits points of is bounded

Theorem.

be a bounded set of real numbers then exist iff

Definition (properly diverges).

note the definition later