Why does sup inf (ak × bk) ≥ sup inf ak × sup inf bk?

[usernot]

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

 

[usernot]

Canany mentor delete this post? I just discovered how it works and i should be fine