I'm reading the proof that $\inf \left\{\exp((-1)^nn)\right\} = 0$ where the solution I'm reading says that we can choose $n=2 \max\left\{ \lceil \log(1/\varepsilon)\rceil,1 \right\}+1$ then $\exp((-1)^nn)< \varepsilon $ to show that any $\varepsilon > 0$ is not a lower bound for the sequence. I'm confused by where the $1$ in $\max\left\{ \lceil \log(1/\varepsilon)\rceil,1 \right\}$ comes from. In my mind, we could choose, say, $n=2 \lceil \log(1/ \varepsilon)\rceil+1$. So clearly my understanding is not right here.
So where does it come from? Thanks in advance.
For $\epsilon > 1$, $\log(1/\epsilon)$ is negative, which could produce a nonsensical result. But with the max in place, then for large $\epsilon$, we choose $n=3$ and are fine since $e^{-3} < \epsilon$.