# A (Faulty) Proof that $(-1)^{n}$ is Monotone

A weird non-proof involving induction that I stumbled upon during my Real Analysis homework. Neither me or my professor could find what was wrong with it.

Theorem: Define the sequence $$a_n$$ as a recursive sequence such that $$a_1 = -1$$ and $$a_{n+1} = -a_n$$. The sequence $$a_n$$ is monotone increasing.

Proof: Proof by induction.

The base case is trivial, as clearly $$a_1=-1<1=a_2$$, so the condition $$a_{n-1} is satisfied for at least one $$n$$.

For the induction step, we consider the quotient $$\frac{a_{n+1}}{a_n}$$ which by definition of $$a_n$$ is equivalent to $$\frac{-a_{n}}{-a_{n-1}}=\frac{a_{n}}{a_{n-1}}$$. By the induction hypothesis, $$a_{n-1} so $$\frac{a_{n}}{a_{n-1}}>1$$, implying $$\frac{a_{n+1}}{a_n}>1$$, implying $$a_{n} Hence, $$a_n$$ is monotone and increasing $$\blacksquare$$.

Can anyone tell me what's going on here? It feel like it's pretty basic, but I have no idea what I'm missing.

$$a_{n-1} does not imply $${a_n \over a_{n-1}}>1$$ unless $$a_{n-1}$$ is positive.