# Prove by contradiction that if $n$ is a natural number then $n/(n+1) > n/(n+2)$

Prove by contradiction that if $$n$$ is a natural number then $$n/(n+1) > n/(n+2)$$

Since it is a proof by contradiction, I think I start by assuming that $$n/(n+1) < n/(n+2)$$, but then I don't know how to prove that either. The graphs make it clear but I don't know how to proceed to give a contradiction

Given that $$n>0$$, if we assume the contrary, i.e., $$\frac n {n+1} \le \frac n {n+2},$$ then we could divide both sides by $$n$$ and cross-multiply to get $$n+2 \le n+1.$$
Subtract $$n$$ from both sides, and there is clearly a contradiction.
• Alternatively, I could have said to subtract $n+1$ from both sides – J. W. Tanner Feb 7 '19 at 23:31
If possible, let $$\frac n {n+1} \leq \frac n {n+2}$$. This gives $$n(n+2) \leq (n+1)n$$ or $$n^{2}+2n\leq n^{2}+n$$. This means $$2n \leq n$$ or $$n \leq 0$$ which is a contradiction. Hence $$\frac n {n+1} > \frac n {n+2}$$.