Conditions for a real number not to be a limit point

  • Thread starter monkeybird
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  • #1

Homework Statement


Given a sequence <xn>n of real numbers.

Give the conditions for a real number a not to be a limit point of the sequence. (lim xn not equal to a.)

The Attempt at a Solution


I'm really not sure if this is the whole answer or if it's only a part of it:

For all e>0 there exists an n that belongs to the real numbers s.t. |xn - a| >= e.


Is there more to this or do I have it correct?
 

Answers and Replies

  • #2
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Consider a closed interval of the form [a,b] where a,b are some reals. Every point in the set is a limit point because for some x in [a,b], the sequence x + 1/n converges to x as n approaches infinite.
 
  • #3
HallsofIvy
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Homework Statement


Given a sequence <xn>n of real numbers.

Give the conditions for a real number a not to be a limit point of the sequence. (lim xn not equal to a.)

The Attempt at a Solution


I'm really not sure if this is the whole answer or if it's only a part of it:

For all e>0 there exists an n that belongs to the real numbers s.t. |xn - a| >= e.


Is there more to this or do I have it correct?
It's not a matter of "more to this", that's not at all correct. For example, the sequence {(-1)n} does not converge to 0 (it doesn't converge at all) but if [itex]\epsilon= 2[/itex] there is NO n such that |(-1)n- 0|= 1> [itex]\epsilon[/itex]. On the other hand, {1/n} converges to 0 but if [itex]\epsilon= 1/4[/itex], for n= 2, |1/2- 0|= 1/2> 1/4.

The definition of "{an} converges to a" is "For all [itex]\epsilon> 0[/itex], there exist N> 0 such that if n> N, |an- a|< [itex]\epsilon[/itex]". The opposite of that is "There exists [itex]\epsilon> 0[/itex] such that for all N, there exist n> N such that |an-a|> [itex]\epsilon[/itex]. Do you see how, in taking the "opposite", "for all" changes to "there exist" and vice-versa?
 

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