you have a typo here. the red n should be a 1 i think

lets say they said n>=3, what happens if n = 3? division by zero

So far, this makes sense. But then they choose to pull out

n^2/[(n-1)(n-2)(n-3)] for n>=4 and call it Cn, so that 0<An<=Cn. Then Squeeze Theorem can be used.

So why did they choose n>=4? Why not n>=3 or n>=5? Is there another way to do this?

they chose the range of n so as to not cause problems with Cn, n <= 3, we have division by zero due to one of the factors (n - 1), (n - 2), or (n - 3)