This was a step glossed over in a proof in the convergence of $\displaystyle a_n=(1+\frac{1}{n})^n$ that I can't seem to understand where certain terms are appearing from.

The identity is

$\displaystyle \dfrac{n(n-1)(n-2)\cdots(n-k+1)}{n^k}=\left(1-\dfrac{1}{n}\right)\left(1-\dfrac{2}{n}\right) \cdots \left(1-\dfrac{k-1}{n}\right)$

I was able to get something similar by dividing $\displaystyle k$ $\displaystyle n$s on the left hand side acros the 0th term $\displaystyle (n-0)$ to the $\displaystyle (k-1)$st term but that still leaves the $\displaystyle k$th and $\displaystyle (k+1)$st term untouched and with a final term not in the form of the right hand side. It also leaves $\displaystyle (n-k)(n-k+1)$in the left hand product.

Just to illustrate what I mean I will take the case of $\displaystyle k=4$

$\displaystyle \dfrac{n(n-1)(n-2)(n-3)(n-4)(n-5)}{n*n*n*n}=\left(\dfrac{n}{n}\right)\left(1-\dfrac{1}{n}\right)\left(1-\dfrac{2}{n}\right)\left(1-\dfrac{3}{n}\right)(n-4)(n-5)$

Something tells me I'm making a wrong assumption somewhere or it's something dumb and obvious but I can't seem to get how this will end up in the form on the lhs.

Any help would be appreciated. Thank you for your time.