How to find out the transfer function of a FIR filter?
\begin{align}
y[n] &amp;= h[n]  * x[n]\\
&amp;= h[0]x[n] + h[1]x[n-1] + \ldots + h[N-2]x[n-(N-2)] + h[N-1]x[n-(N-1)]\\
&amp;= a^{0}x[n] +a^{1}x[n-1] + \ldots + a^{N-2} x[n-(N-2)] + a^{N-1} x[n-(N-1)]
\end{align}
So

$$\mathcal Z\{y[n]\} = Y(z) = X(z)\left(1 + az^{-1} + ... + a^{N-2}z^{-(N-2)} + a^{N-1}z^{-(N-1)}\right)$$

and

$$H(z) = \dfrac{Y(z)}{X(z)} = 1 + az^{-1} + ... + a^{N-2}z^{-(N-2)} + a^{N-1}z^{-(N-1)}$$

For $a$ finite, $H(z)$ is finite and thus stable for finite input values.