Finding the product $\prod_{k=2}^n \left(1 - \frac{1}{k}\right)$ using Induction

Question

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

Answer 1

General induction proofs need to be structured like this:

• Base Case: Prove your statement for $n=2$. Plug in and validate this manually.
• Inductive Step: Assume your statement holds fon some $n=N$, and prove it holds for $n = N+1$. To do that, note that $$ \prod_{k=2}^{n+1} f(k) = f(n+1) \prod_{k=2}^n f(k) $$

Answer 2

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 $$ \left(1 - \frac{1}{m+1} \right) \cdot \prod_{k = 1}^m \left( 1 - \frac{1}{k}\right) = \left(1 - \frac{1}{m+1} \right) \cdot \frac{1}{m} = \left(\frac{1}{m} - \frac{1}{(m + 1)m} \right) = \frac{(m+1) - 1}{m(m+1)} = \frac{1}{m+1}. $$