Proof that $1$ is an upper bound of $a_{n}:=(-1)^n$ using induction

Question

I would like to prove that $1$ is an upper bound of $a_{n}:=(-1)^n$ using induction. I am stuck in the inductive step, namely, if $1\geq (-1)^n$, then $1\geq (-1)^{n+1}$.

I know that $1\geq (-1)^n \Leftrightarrow (-1)\times 1\geq (-1)\times (-1)^n \Leftrightarrow 1\geq -(-1)^{n+1}$. But that seems to go nowhere.

Edit: typo: it has to be $1\geq (-1)^n \Leftrightarrow (-1)\times 1\leq (-1)\times (-1)^n \Leftrightarrow 1\geq -(-1)^{n+1}$.

Answer 1

One way to go is to make your inductive statement be that $a_{n+1} \in \{-1,1\}$, a set that is clearly upper bounded by $1$.