What is the definition of supremum and how can it be used to compare sequences?

[Somefantastik]

My question involves supremums and their implications:

say I have the sequences $\left\{x_{k}\right\}_{k=1}^{\infty}$ and $\left\{y_{k}\right\}_{k=1}^{\infty}$

and I know $sup \left\{x_{k}:k\in N \right\} \leq sup \left\{y_{k}:k\in N \right\}$

What can I say about the sequence $\left\{x_{k}\right\}_{k=1}^{\infty}$?

Can anyone suggest a text or website that goes into detail about supremums and how to use them? All my Analysis books just announce the definition of a supremum and move on, but I'd like to see a little more exposition, especially with what can be done with two sets correlated by their sups or infs as above.

 

[micromass]

Well, you can probably say that for all $x_n$, there exists an $y_m$ such that $x_n\leq y_m$. Is this what you wanted??

Just read all the proofs involving sups and infs, you'll see soon enough what you can do with them

 

[solakis]

My question involves supremums and their implications:

say I have the sequences $\left\{x_{k}\right\}_{k=1}^{\infty}$ and $\left\{y_{k}\right\}_{k=1}^{\infty}$

and I know $sup \left\{x_{k}:k\in N \right\} \leq sup \left\{y_{k}:k\in N \right\}$

What can I say about the sequence $\left\{x_{k}\right\}_{k=1}^{\infty}$?

Can anyone suggest a text or website that goes into detail about supremums and how to use them? All my Analysis books just announce the definition of a supremum and move on, but I'd like to see a little more exposition, especially with what can be done with two sets correlated by their sups or infs as above.

Nothing can be said about the sequences ,because they may both be vibrating over different fixed points without converging to any limits .

If however both,for example are increasing to their least upper bounds then we can say:

$lim_{k\rightarrow\infty} x_{k}\leq lim_{k\rightarrow\infty}y_{k}$

TO get an idea as to the richness of the unexplorable fields concerning supremums and infemums,consider the following.

Let X,Y be a set of positive real Nos bounded from above. Then define : X+Y, X.Y , $\sqrt X$ , $X^n$ ,nεN ,and try to find their supremums.

Not to forget of course that the whole theory of integral calculus is based on the definitions of supremums and infemums.

Of the outermost importance is a correct and proper definition .