Every convergent sequence has a monotoic subsequence

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Homework Statement


Prove that every convergent sequence has a monotone subsequence.

Homework Equations




The Attempt at a Solution


Define ##L## to be the limit of ##(a_n)##. Then every ##\epsilon##-ball about L contains infinitely many points. Note that ##(L, \infty)## or ##(-\infty, L)## (or both) has infinitely many elements. Suppose that ##(L, \infty)## has infinitely many elements. Choose ##n_1## such that ##a_{n_1} \in (L, \infty)##. Choose ##n_2 \in (L,a_{n_1})##. In general, choose ##n_{k+1}## such that ##a_{n_{k+1}} \in (L, a_{n_{k}})##. This assignment can always be made since there are infinitely many elements of ##(a_n)## in ##(L, a_{n_{k}})## to choose from such that ##a_{n_{k+1}} \in (L, a_{n_{k}})##.
 

Answers and Replies

  • #2
FactChecker
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Do you have a question about this? I think it is a good start and only needs more supporting detailed statements to make it a rigorous proof.
 
  • #3
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Do you have a question about this? I think it is a good start and only needs more supporting detailed statements to make it a rigorous proof.
I guess my question then would be what supporting detailed statements do I need? As of now this is the best I can do, so I'm trying to see what I'm missing in terms of rigor.
 
  • #4
FactChecker
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Just add more description. The words do not cost you anything. What about the lower interval? Is the sequence increasing or decreasing? Monitone because?
 
  • #5
Stephen Tashi
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Then every ##\epsilon##-ball about L contains infinitely many points.
It need not contain infinitely many distinct points. The sequence 2,2,2,2,.... converges to 2.
 
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