Help with Math Proof: Bounded Sequence (xn), (yn) & Limsup (xn)

[SR2]

Let (xn) be a bounded sequence. Denote yn=sup{xm : m is greater or equal to n}. Prove

(a) (yn) is a decreasing and bounded below.

Thus, by MCT, there exists a lim(yn)=inf(yn). this limit is called limsup(xn).

(b) Is (yn) necessarily subsequence of (xn)? if so, argue why, if not, give an example where (yn) is not a subsequence of (xn).

(c) Is it true that every bounded sequence (xn) has a subsequence convergent to limsup(xn)?. If so prove it (This would provide an alternative proof for BWT!)

I feel quite clueless- what elements are there in yn.
- how do you prove that yn is decreasing if xn is increasing?
-is xm a subsequence of xn?

-also it would help if I could get a rough idea of how to prove part a and c.

 

[micromass]

Hi SR2!

With these kind of questions (and with every math question actually), I find it best to start of with some examples. So, let's say I give you the following sequences:

$$(-1)^n,~\frac{1}{n},~\frac{(-1)^n}{n},~-\frac{1}{n}$$

can you calculate the correspond sequence $(y_n)$ for me and can you calculate the limsup for me?