Every convergent sequence has a monotoic subsequence

[Mr Davis 97]

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 $a_{n_{k+1}} \in (L, a_{n_{k}})$.

 

[FactChecker]

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.

 

[Mr Davis 97]

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]

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]

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.