I try to prove that:
Statement (S): $\frac{1}{n-1}(1-\frac{1}{n}) = \frac{1}{n}$
by induction for all $\forall n \in \mathbb{N} \setminus$ {$ 1 $}.
Here is my solution:
Base case: S(2) $ = \frac{1}{2-1} (1 - \frac{1}{2}) = \frac{1}{1} - \frac{1}{2} = \frac{1}{2}$, which is true.
Assuming S(n) is true, then: S(n+1) $ = \frac{1}{n} (1 - \frac{1}{n+1}) = \frac{1}{n}(\frac{n+1}{n+1} - \frac{1}{n+1}) = \frac{1}{n}(\frac{n}{n+1}) = \frac{1}{n+1}$,
which concludes the proof. Is the solution correct, or is there an error in my argument.