Does convergence imply uniform convergence in this example?

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:

1. ${y_k} rightarrow 0$ for all ${x_k}in[0~1]^mathbb{N}$

2. $y_k=f(y_{k-1},x_k)$ for all $kinmathbb{N}$, 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?