  Let $\beta>1$. For $x \in [0,\infty)$, we have so-called the
$\beta$-expansion of $x$ in base $\beta$ as follows: $$x= \sum_{j \leq k}
x_{j}\beta^{j} = x_{k}\beta^{k}+ \cdots + x_{1}\beta+x_{0}+x_{-1}\beta^{-1} +
x_{-2}\beta^{-2} + \cdots$$ where $k \in \mathbb{Z}$, $\beta^{k} \leq x <
\beta^{k+1}$, $x_{j} \in \mathbb{Z} \cap [0,\beta)$ for all $j \leq k$ and
$\sum_{j \leq n}x_{j}\beta^{j}<\beta^{n+1}$ for all $n \leq k$. In this paper,
we give a sufficient condition (for $\beta$) such that each element of
$\mathbb{N}$ has the finite beta-expansion in base $\beta$. Moreover we also
find a $\beta$ with this finiteness property which does not have positive
finiteness property.
