I want to prove the following statement which seems like an obvious statement but I am having trouble proving it. Let $ \mathcal{C} \subseteq \mathbb{R}^2$ be a set diffeomorphic to $ (0,1)^2$ . Let us denote $ $ \mathcal{C}’ = \cup_{ \mathbf{x} \in \mathcal{C} } \ (\mathbf{x} + (- \eta, \eta)^2). $ $ Let $ \mathbf{y} \in \mathbb{R}^2$ . Suppose $ $ (\mathbf{y} + [-10 \eta, 10 \eta]^2) \subseteq \mathcal{C}’. $ $ Can I then conclude that $ $ (\mathbf{y} + [- \eta, \eta]^2) \subseteq \mathcal{C}. $ $

I am wondering how I can prove this statement. Even though it seems it should be obvious, I am struggling to prove it… I would greatly appreciate any comments or suggestions.