Finding the product $\prod_{k=2}^n \left(1 - \frac{1}{k}\right)$ using Induction I'm trying to figure out how to prove by induction the following statement:
$$
\prod_{k=2}^n \left(1 - \frac{1}{k}\right) = \frac{1}{n}.
$$

Inductive hypothesis:
Assume that 
$$
\prod_{k = 1}^m \left( 1 - \frac{1}{k}\right) = \frac{1}{m}
$$
holds true for some $m$. 
Then we compute
$$
\left(1 - \frac{1}{m+1} \right) \cdot \prod_{k = 1}^m \left( 1 - \frac{1}{k}\right) = \left(1 - \frac{1}{m+1} \right) \cdot \frac{1}{m} = \left(\frac{1}{m} - \frac{1}{(m + 1)m} \right) = \frac{(m+1) - 1}{m(m+1)} = \frac{1}{m+1}.
$$
