How can the lim inf and lim sup be used to solve this mystery?

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In summary, the problem involves demonstrating that the product of the limit inferior of two sequences is less than or equal to the limit inferior of their product. After some algebraic manipulation and using the fact that the supremum of the infimum of a set is equal to the limit inferior, the problem can be solved.
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Homework Statement


Demonstrate lim inf an × lim inf bn ≤ lim inf (an × bn)

Homework Equations

The Attempt at a Solution


Fixxed n € N , n≤ k , inf ak ≤ ak , inf bk≤ bk =》inf ak × inf bk ≤ (ak × bk) . Being inf ak × inf bk ≤ any element from {ak} × {bk} it`s logical that inf ( ak × bk ) ≥ inf ak × inf bk... the part i don't get it's the following: the teacher suddently writes : sup inf ( ak × bk ) ≥ sup inf ak × sup inf bk.. the part after this is much easier since sup inf ak = lim inf an , sup inf bk = lim inf bk and sup inf ( ak × bk ) = lim inf ( ak×bk)
Can anyone solve this mistery? Many Thanks in advance
 
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Canany mentor delete this post? I just discovered how it works and i should be fine
 

What is lim inf and lim sup?

Lim inf (limit inferior) and lim sup (limit superior) are two important concepts in mathematical analysis. They are used to describe the behavior of a sequence or function as the input variable approaches a certain value.

How are lim inf and lim sup calculated?

Lim inf and lim sup are calculated by finding the infimum (greatest lower bound) and supremum (least upper bound) of a sequence or function, respectively. In other words, they are the smallest and largest possible values that the sequence or function can approach as the input variable approaches a certain value.

What is the difference between lim inf and lim sup?

The main difference between lim inf and lim sup is that while lim inf is the smallest possible value that a sequence or function can approach, lim sup is the largest possible value. Another way to think about it is that lim inf is a lower bound and lim sup is an upper bound for the sequence or function.

How are lim inf and lim sup used in real-world applications?

Lim inf and lim sup are used in many areas of mathematics, including calculus, analysis, and probability. They are also commonly used in areas such as physics, engineering, and economics to describe the behavior of systems or processes as a variable changes.

Can lim inf or lim sup be infinite?

Yes, either lim inf or lim sup can be infinite in certain cases. For example, if a sequence or function has no upper or lower bound, then lim inf or lim sup will be infinite. Similarly, if a sequence or function approaches infinity as the input variable approaches a certain value, then lim inf or lim sup will also be infinite.

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