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

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The discussion centers on demonstrating the inequality lim inf an × lim inf bn ≤ lim inf (an × bn). The user initially attempts to prove the relationship by establishing that the infimum of products is greater than or equal to the product of infima. Confusion arises when the teacher introduces the concept of supremum in relation to the infimum of products. The user later realizes the solution and expresses gratitude, indicating that they no longer need assistance. The thread highlights the complexities of limits and infimum/supremum in mathematical analysis.
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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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