There is no smallest integer \(k\) because

\begin{equation*} \forall k \in \mathbb{Z} \quad \exists \ell \in \mathbb{Z} \quad (\ell < k), \end{equation*}

that is

for every integer \(k\),
there exists some integer \(\ell\)
smaller than \(k\),

such as \(\ell = k - 1\).