Proving properties of lim sup. Good attempt at proof

In summary, the statement 'if (cn) and (dn) are bounded sequences of positive real numbers, then lim sup (cndn) = (lim sup cn)(lim sup dn) for all n in the positive reals' is true. This is because bounded sequences are known to converge, and by the theorem lim(cndn)=lim cnlim dn, we can say that lim sup (cndn) = (lim sup cn)(lim sup dn).
  • #1
tamintl
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Homework Statement


Is the following statement true or false: 'if (cn) and (dn) are bounded sequences of positive real numbers then:

lim sup (cndn) = (lim sup cn)(lim sup dn)

Homework Equations


The Attempt at a Solution


for all n in the positive reals. cn and dn are bounded.

Since cn and dn are bounded we know they converge.

Hence, by the theorem lim(cndn)=lim cnlim dn we can say that lim(cn)=lim sup (cn) and lim(dn)=lim sup (dn)

Hence, lim sup (cndn) = (lim sup cn)(lim sup dn)

HENCE, TRUE!

******I am not sure about line 2 where I say bounded => converge. I know that you can say that converge means bounded but I don't know if I can do the reverse.*******
 
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  • #2
tamintl said:
******I am not sure about line 2 where I say bounded => converge. I know that you can say that converge means bounded but I don't know if I can do the reverse.*******

You can't. You can say that a bounded monotone sequence converges and that a bounded sequence has a convergent subsequence. But a bounded sequence need not converge. You should be able to come up with a pretty simple example of a bounded sequence that does not converge (at least now that you know there is one). In the process of doing so, you may stumble across the answer to your problem.
 

1. What is lim sup?

Lim sup, or limit superior, is a mathematical concept that represents the largest limit point of a sequence or a function. It is defined as the supremum of the set of all accumulation points of the sequence or function.

2. How is lim sup different from lim inf?

Lim inf, or limit inferior, is the opposite of lim sup and represents the smallest limit point of a sequence or function. Unlike lim sup, which looks at the "highest" limit point, lim inf looks at the "lowest" limit point.

3. How can I prove properties of lim sup?

To prove properties of lim sup, you can use the definition of lim sup and apply various mathematical techniques such as inequalities, limits, and convergence tests. It is also helpful to have a clear understanding of the properties of sequences and functions.

4. Why is proving properties of lim sup important?

Proving properties of lim sup is important because it allows us to understand the behavior of a sequence or function at its largest limit point. This information can be crucial in various mathematical applications, such as determining the convergence of a series or finding the maximum value of a function.

5. Is there a general approach for proving properties of lim sup?

There is no one specific approach for proving properties of lim sup, as it depends on the specific property being proven. However, some common techniques include using the definition of lim sup, applying known properties of limits and sequences, and using mathematical induction.

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