Transient response of system with single pole $0 \le p < 1$
If you have a single real-valued pole, you get a term $k\,p^{(n-n_0)}u[n-n_0],n\ge 0$, with some constant $k$ and some delay $n_0\ge 0$ in the system's impulse response ($u[n]$ is the unit step). Clearly, if $|p|\ge 1$ the transient will never settle. If $-1&lt;p&lt;0$, the transient will alternate in sign (i.e., "oscillate"), like e.g.,

$$\left(-\frac12\right)^nu[n]=\left\{1,-\frac12,\frac14,-\frac18,\dots\right\}\tag{1}$$

So for a transient that decays (i.e., for the system to be stable) and that does not oscillate you need $0\le p&lt;1$.

Note that if you only have one real-valued pole $p$, stability implies $$-1&lt;p&lt;1\tag{2}$$ If you exclude systems with oscillating impulse responses (i.e., with a negative pole), you're left with the requirement $0\le p&lt;1$. But, as you've noticed, this requirement is not only dictated by stability (which only requires $(2)$), but also by the requirement of having no oscillations.