Suppose we have a sequence $\{u_t \}_{t \in \mathbb{N}}$ given by the recurrence relation $$u_{t+1} = q_0 u_t + q_1 u_{t-1} + \dots + q_p u_{t-p} + \epsilon, \quad \epsilon >0$$ where $p \in \mathbb{N}$ and $q_1, \dots, q_p$ are such that the roots $r_0,r_1, \dots, r_p$ of the associated homogeneous characteristic equation $$r^{p+1} - q_0 r^p - q_1 r^{p-1}- \dots - q_p = 0 $$ lie inside the unit circle and the sequence is strictly positive.
We can render the sequence homogeneous by settting $u^* = \epsilon / (1 - q_0 - q_1 - \dots -q_p) $ and noticing that $$(u_{t+1} -u^*) = q_0 (u_t - u^*) + q_1 (u_{t-1} - u^*) + \dots + q_p (u_{t-p} - u^*) $$ It is known that the solution to this recurrence will have the form $$u_{t+1} - u^* = a_0 r_0^{t+1}+a_1 r_1^{t+1}+ \dots + a_p r_p^{t+1}$$ where $a_0,a_1, \dots, a_p$ are real numbers and $r_0, r_1, \dots , r_p$ are the roots of the characteristic equation. Taking the limit for $t \rightarrow \infty$ and recalling that the roots where assumed inside the unit circle we obtain that $$\lim_{t \rightarrow \infty} u_{t+1 } = u^*$$ but (it seems to me) $u^*$ can be negative and the sequence was taken strictly positive. Where is the mistake?
Edit: The Eneström–Kakeya theorem (see this link) seems to imply that we can construct a polynomial with all roots inside the unit circle and such that $u^* = \epsilon / (1 - q_0 - q_1 - \dots -q_p) $ is negative. So I still have a doubt if this derivation is correct, thus I am putting a bounty on the question.
Under the given assumptions, we can show that $u^*>0$. We can observe that $$ 1-q_0-q_1-\cdots -q_p = p(1) $$ where $$ p(r) = r^{p+1}-q_0r^p-q_1r^{p-1}-\cdots -q_p $$ is the characteristic polynomial. Note that $\lim\limits_{r\to\infty}p(r) =\infty$. If $p(1)< 0$, then by the intermediate value theorem, there exists $r_0\in(1,\infty)$ such that $p(r_0)=0$. Since we are assuming that every root of $p(r)$ lies in the open unit disk, this leads to a contradiction. Thus we have $p(1)> 0$ as $p$ cannot have $1$ as a root. As a result, we get $u^* =\frac{\epsilon}{p(1)}>0$.