Consider a map \begin{align} F:&[0~1]^\mathbb{N}\rightarrow \mathbb{R}^\mathbb{N}\\ &\{x_k\}\rightarrow \{y_k\}=F\{x_k\} \end{align} with the following properties:
$\{y_k\} \rightarrow 0$ for all $\{x_k\}\in[0~1]^\mathbb{N}$
$y_k=f(y_{k-1},x_k)$ for all $k\in\mathbb{N}$, where $f(\cdot,\cdot)$ is a continuous function and $y_0$ is a given constant.
Can we guarantee all the possible sequences $\{y_k\}$ converge uniformly to zero?