Sorry, I didn't read the title.
Okay, we have the "base" case:
So this works for n = 2.
Let's assume this works for some n = k, that is that
Let's see what the n = k + 1 case says:
By hypothesis, the product of all but the last factor is simply 1/k, so:
Simplifying a bit:
as required.
So it is true for n = 2, thus it is true for n = 3, 4, ...
-Dan