Sup of a sequence

Bachelier

Is my understanding of the concept of ##\underset{n}{Sup} \ S_n## correct?

for instance, given the sequence:

##{S_n} = sin(\frac{n \pi}{2}). \frac{n+2}{2 n}##

Then

##\underset{1}{Sup} \ S_n \ = \ \frac{3}{2}##

##\underset{10}{Sup} \ S_n \ = 0##

##\underset{k≥n}{Sup} \ S_n \ = \ \frac{1}{2}##

I am trying to understand the part when we say ##\underset{n}{Sup} \ S_n##, what does it mean? Thanks

rasmhop

I doesn't look like you really understand sup. The n below sup is meant as a variable, and sometimes we have restrictions such as
[tex]\sup_{n\geq 5} x_n[/tex]
which means we consider sup of the sequence [itex]x_5,x_6,x_7,\ldots[/itex]. Therefore your first two statements do not seem to make sense (at least with standard notation). One could argue that formally we should have
[tex]\sup_{n=1} S_n = S_1 = 3/2[/tex]
but I have never seen anyone use sup for a single value since it is always equal to just the value.

To understand sup you should first understand max of a sequence. Some sequences have a maximum value (i.e. you can find a fixed k such that [itex]x_k \geq x_n[/itex] for all n). When this is the case then
[tex]\sup_{n} x_n = \max_n x_n = x_k[/tex]
However max does not always make sense. For instance consider the sequence
[tex](0, 1/2, 2/3, 3/4, \ldots)[/tex]
This sequence does not have a maximum because every element gets bigger. However we can see that the entries get arbitrarily close to 1. This is precisely what sup is. sup of a sequence is simply the smallest number that is still greater than every single element. It is a way to define something like max, but for all sequences.

The sequence you have given actually has a maximum value so the sup is just that value. I'm not going to point out what the value is because you should be able to see it yourself and in case this is homework. I can at least tell you that it is not 1/2, you can find elements that are greater than 1/2.

Bacle2

No Sup for you! NEXT.

Bacle2

Sup is also called least-upper bound. The sup is the least of all upper bounds. A simple

example : take the interval (0,1) --take it, please!. No, sorry, now, what are the

possible upper bounds for the set of all x's in (0,1)? Well, 2 is an upper bound, so is 3,

and so is any number larger than 3. But which is the least among all upper bounds?

It is 1. It is a little involved, but not too hard to show this.

Now, you need to do the same for your collection of objects Sn . Notice, Sn is

the set {S1,S2,....} . Out of all the numerical values of Sn, can you think of the

least real number that is larger than all the Sn's? If you can figure out, a proof

should not be too far.
Sn

Bachelier

Thank you ramhop. It is not homework. I took the example from a youtube video (see link below)

(Warning: Sie haben Deutsch zu sprechen) :)




I understand the Least Upper Bound property, but what confuses me is the "limsup" notation. I kind of understand it as the limit of the sup of the tail of a certain series.
i.e. ##T_n = \left\{{S_k | k ≥ n}\right\}## for some sequence ##(S_n)##
Please feel free to expand on this!

The examples I posted were a meek attempt to try and understand some notations I found in the wikipedia article about the subject on

http://en.wikipedia.org/wiki/Limit_s...limit_inferior

for instance see the image below:


especially the inequality:


## \underset{n}{Inf} \ S_n \ ≤ \underset{n→∞}{limInf} \ S_n ≤ \underset{n→∞}{limSup} \ S_n ≤ \underset{n}{Sup} \ S_n##

Now how do I look at: ##\underset{n}{Sup} \ S_n## ? should it be considered as the sup at a certain n or as the sup of all ##S_k## s.t. ##k ≥ n## or the sup of the whole sequence? I think the correct answer is the last one.

Bachelier

Good one. I prefer Salads anyway.

Bachelier

Although not the original intention of my question, I will go ahead and determine the Sup of this sequence. Since ##(S_n)## is monotonically "Fallend" non-increasing, ##S_1 = 1.5## is the sup.

Bacle2

Well, but , under the standard notation, # \underset{n}{Sup} \ S_n # is:

Sup{S₁,S₂,......, S_n,......}

Bachelier

Thanks Bacle. I am now more familiar with this concept. I just wanted to make sure I am familiar with the notation I saw on Wikipedia and that I understood it correctly.