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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 $a_{n-1}<a_{n}$ 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, $a_{n-1}<a_{n}$ so $\frac{a_{n}}{a_{n-1}}>1$, implying $\frac{a_{n+1}}{a_n}>1$, implying $a_{n}<a_{n+1}.$ 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.

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1 Answer 1

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

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