Closure of a set of complex numbers

Question

I need to solve one task from complex analysis. Let $ A\subset \mathbb{C} $ be a bounded set in space $ (\mathbb{C},d) $ where $ d $ is the Euclidean metric on the complex plane $ \mathbb{C}. $ Show that the closure of the set $ A $ in space $ (\mathbb{C},d) $ is the same as the closure of the set $ A $ in space $ (\mathbb{C},\rho), $ where $ \rho $ is the metric given by $$ \rho (z_1,z_2)=\frac{|z_1-z_2|}{\sqrt{1+|z_1|^2}\cdot \sqrt{1+|z_2|^2}}$$ for $ z_1, z_2\in \mathbb{C}. $ I have shown that $ d $ and $ \rho $ are strongly equivalent metrics, but does that imply equality of mentioned closures? I would really appreciate your help in resolving this problem.

Answer 1

When two metrics are equivalent, they induce the same topology, and therefore the closure of a set with respect to one of them is its closure with respect to the other one.