Product by induction

Base case. $n=1$: $1+1=2$. ✓

Inductive step. Assume the product equals $n+1$. Then
$$\prod_{k=1}^{n+1}\left(1+\frac1k\right)=(n+1)\cdot\left(1+\frac{1}{n+1}\right)=(n+1)\cdot\frac{n+2}{n+1}=n+2.$$

Conclusion. The product equals $n+1$ for all $n\ge1$.