# Does convergence imply uniform convergence in this example?

Mathematics Asked by Daniel Huff on January 1, 2022

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?

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