Proving limsup Inequality for Bounded Sequences

[steelphantom]

Homework Statement

Let (s_n) and (t_n) be bounded sequences of nonnegative numbers. Prove that limsup(s_n*t_n) <= (limsup(s_n))*(limsup(t_n)).

Homework Equations

The Attempt at a Solution

I know that (s_n) and (t_n) have convergent subsequences, but that's about it. Where could I go from here? Thanks!

 

[quasar987]

What you could do is suppose the contrary; i.e. suppose limsup(s_n*t_n) > (limsup(s_n))*(limsup(t_n)), and then find a counterexemple to this by choosing the sequences s_n and t_n adequately.

 

[steelphantom]

What you could do is suppose the contrary; i.e. suppose limsup(s_n*t_n) > (limsup(s_n))*(limsup(t_n)), and then find a counterexemple to this by choosing the sequences s_n and t_n adequately.

Ok, I can do that. So, basically I'm saying: there exists x such that not A fails => A is true for all x. Is this the idea? Thanks!

 

[quasar987]

Yep.

 

[Vid]

When you negate a for every you get a there exists.

Not A should say:

There exists two bounded non-negative sequences s_n, t_n such that
limsup(s_n*t_n) > limsup(s_n)*limsup(t_n).


Finding a sequence that doesn't fit that doesn't prove a damn.

 

[quasar987]

Oh yes, Vid is right. What I suggested doesn't work.

 

[steelphantom]

Ok, this just doesn't seem right to me. Isn't this just equivalent to coming up with an example of what I want to prove, and not really proving it?

For example, what if I tried to "prove" x + 1 > 0 for all x in R. If I assume the converse, i.e. x + 1 <= 0 for all x in R, and I take x to be 1, I get x + 1 = 2 > 0. So the converse is false, but the original statement is obviously not true.

Edit: I was posting this while Vid posted. That's what I was suspecting. How can I try to prove this then?

 

[Vid]

You could still assume not and try and derive a contradiction.

 

[steelphantom]

Ok, well intuitively, I see why this is true, because limsup(s_n) is the biggest number in S = {limits of all subsequences of s_n} and limsup(t_n) is the biggest number in T = {limits of all subsequences of t_n}. So multiplied together, limsup(s_n)*limsup(t_n) has to be >= limsup(s_n*t_n) because s_n*t_n can't ever be bigger than sup(T)*sup(S).

I think I know what's going on. But I'm really not sure on how to prove it, either by deriving a contradiction or otherwise. Thanks for your help.

 

[Vid]

Use the fact that they are bounded sequences. If |s_n|<=M and |t_n| <=N. Since everything is positive |s_n||t_n| <= MN. If limsup(s_n*t_n) > limsup(s_n)*limsup(t_n)...

Can you get it from here?

 

[steelphantom]

Use the fact that they are bounded sequences. If |s_n|<=M and |t_n| <=N. Since everything is positive |s_n||t_n| <= MN. If limsup(s_n*t_n) > limsup(s_n)*limsup(t_n)...

Can you get it from here?

Well, can I say that limsup(s_n) <= s_n <= M, and also that limsup(t_n) <= t_n <= N? So limsup(s_n)limsup(t_n) <= MN. And limsup(s_n*t_n) <= (s_n*t_n) <= MN. Then we have MN >= limsup(s_n*t_n) > limsup(s_n)limsup(t_n). But this is a contradiction, because we could have limsup(s_n)limsup(t_n) = MN.

Is this right?

 

[quasar987]

The idea is sound but limsup(s_n) <= s_n is not right.

What you want to invoke is the fact that s_n is bounded by M, and thus so is every subsequence of s_n, and thus so is the limit of any convergent subsequence of s_n, and thus so is limsup(s_n) (as the sup over all these limits of convegent subsequences)

 

[quasar987]

I'm talking too fast again.

limsup(s_n)limsup(t_n) < MN

is not a contradiction with

limsup(s_n)limsup(t_n) <= MN.

 

[steelphantom]

Ok, so I can say that since (s_n) <= M, then any subsequence (s_n_k) <= M. Since s_n is bounded, it has a convergent subsequence. If lim(s_n_k) = s, then s <= M. limsup(s_n) = sup(S), where S = {limits of all subsequences of s_n}, so limsup(s_n) <= M. Similarly, limsup(t_n) <= N.

Then since u_n = (s_n*t_n) <= MN, any subsequence of u_n <= MN. Since u_n is bounded, it has a convergent subsequence. If lim(u_n_k) = u, then u <=MN. limsup(u_n) = sup(U), where U = {limits of all subsequences of u_n}, so limsup(u_n) <=MN. Then we have MN >= limsup(s_n*t_n) > limsup(s_n)limsup(t_n).

Edit: I just saw your other post, so this isn't right. This is pretty frustrating. It's a pretty easy proof, I know, but I'm just not seeing it. Thanks again for your assistance.

 

[quasar987]

Since I don't know what Vid was hinting at, I can try to guide you through a solution I found if you'd like?

Suppose that for some sequences {s_n}, {t_n}, we have

limsup(s_n*t_n) > (limsup(s_n))*(limsup(t_n))

Let's try to derive a contradiction. Because both sequences are non negative and bounded, then so is s_n*t_n. Set limsup(s_n*t_n) = L (with 0<L<+infinity). This means there exists a subsequence {s_n_k*t_n_k} such that s_n_k*t_n_k-->L.

Now use the fact that {s_n_k} is bounded to deduce that there exists a convergent subsequence s_n_k_l-->s. In particular, s_n_k_l*t_n_k_l-->L , and so

$$\frac{1}{t_n_k_l}=\frac{s_n_k_l}{s_n_k_lt_n_k_l}\rightarrow s/L$$

(note that because L>0, then s_n_k_l*t_n_k_l >0 for l large enough)

This means that t_n_k_l -->L/s. So we can write limsup(s_n)>=s and limsup(t_n)>=L/s, and hence limsup(s_n)*limsup(t_n)>=L, a contradiction. (!)

Surely there is simpler.