Every convergent sequence has a monotoic subsequence

Homework Statement

Prove that every convergent sequence has a monotone subsequence.

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

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FactChecker
Science Advisor
Gold Member
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.

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.

FactChecker
Science Advisor
Gold Member
Just add more description. The words do not cost you anything. What about the lower interval? Is the sequence increasing or decreasing? Monitone because?

Stephen Tashi
Science Advisor
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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