Proof of Unstability

Question

Let assume that the dynamical system $x_{t+1}=f(x_t)$ (with $f(0)=0$) is globally unstable (which means from any initial condition $x\neq0$, the states go to infinity). Can we prove that $x_{t+1}=f(x_t)+y_t$ (where $y_t$ is norm bounded $||y_t||<c$ and is not a function of $x_t$. $y_t$ also changes its vlue over time with some unknown random distribution. So its value is not fixed over time) is unstable for some $x_0 \neq 0$ with $||x_0||\leq c$ for any $c>0$?