Let and be the max. power of prime dividing and respectively , then we need to show that
First note that so
For , we thus have
Let and , then
( since if and now , . )
We can prove by induction , :
It is true when and assume the case is true for any natural number . When we jump to the next case , we have which completes the proof .
It may be enough to note/prove that for any n,m,a (all positive integers).
This can be shown easily using induction on m.
Once this is established use this fact in the series to find the power of a prime in n! i.e. N_p(n) (as per notation used by simplependulum)
This might just give a little simpler proof for the same.