# Thread: Any bounded seq. has a converging subseq. <=> every bounded monotone seq. converges

1. ## Any bounded seq. has a converging subseq. <=> every bounded monotone seq. converges

How would I go about proving such a statement? Any help would be great thanks.

2. Originally Posted by dannyboycurtis
How would I go about proving such a statement? Any help would be great thanks.
First prove this theorem: Every sequence contains a monotone subsequence.
Then prove that Every bounded non-increasing (non-decreasing) sequence converges to its least upper bound (greatest lower bound).

Suppose that $\displaystyle \left( {x_n } \right)$ is a sequence.
Define a set $\displaystyle S = \left\{ {n:\left( {\forall j > n} \right)\left[ {x_j > x_n } \right]} \right\}$.
Take notice of this critical fact: $\displaystyle t \notin S\text{ if and only if }\left( {\exists K > t} \right)\left[ {x_K \leqslant x_t } \right]$.

There are two cases to consider: $\displaystyle S$ is infinite.
In this case there is an increasing subsequence.

If $\displaystyle S$ is finite then there is a non-increasing subsequence.

3. Im still not quite sure how this proves the two statements are equivalent.
Is this proof basically showing that an arbitrary monotone convergent sequence can be shown to be a subsequence of some other bounded sequence?

4. Originally Posted by dannyboycurtis
Im still not quite sure how this proves the two statements are equivalent.
Is this proof basically showing that an arbitrary monotone convergent sequence can be shown to be a subsequence of some other bounded sequence?
Well, that would be trivial! Just add some numbers to your monotone sequence! For example, the monotone sequence $\displaystyle \{x_1, x_2, x_3, ...\}$ is a subsequence of sequence $\displaystyle \{x_1, 1, x_2, 1, x_3, 1, ...\}$

But what you want is the other way- that every sequence contains a monontone subsequence. I'll give you a start: Let S be the set of all indices i such that if j> i, then $\displaystyle a_i< a_j$. In other words, i is in S if and only if $\displaystyle a_i$ is less than all succeeding numbers in the sequence. Now, consider two cases:
1) S is infinite.
2) S is either finite or empty.

5. How could I use the Bolzano Wierstrass theorem for sets to prove this?