Can you guys help me find the error in this. If we start with
$≥1$ and $k$ is an integer and we know that $k > \ln(k)$, then
$$^>$$ $$1>^{−}$$ $$−1<−^{−}$$
But we know that the lowest value of $k$ is one. Plugging this in gives us
$$−1<−^{−1}≤−^{−}$$ Solving for $k$ with the Lambert $W$-function gives us, $(−^{−1})≤−$.
But $(−^{−1})=−1$, so we have $$−1≤−$$ then adding $1$ and $k$ to both sides gives us $≤1$, but we started with $≥1$, so this would seem to indicate that $=1$ exactly. This is clearly false, but I can't find the hole in the logic.