 101
 3
 Problem Statement
 Prove that (0,1) ≈ R.
 Relevant Equations

arctan: R→(π/2,π/2)
arctan(x)=y ⇔ tan(y)=x for x∈R and y∈(π/2,π/2)
I am slightly confused as to how to prove this. I know that two sets are equinumerous if there is a bijection between them. So we are trying to find f: (0,1)→R? It told us that we may assume the inverse tangent function so that would mean
arctan: R→(π/2,π/2). This satisfies arctan(x)=y ⇔ tan(y)=x for x∈R and y∈(π/2,π/2). We were also given that we could assume arctan is a bijective without proof. I am having trouble moving forward from this information and how to construct a proof out of all of these.
Thank you.
arctan: R→(π/2,π/2). This satisfies arctan(x)=y ⇔ tan(y)=x for x∈R and y∈(π/2,π/2). We were also given that we could assume arctan is a bijective without proof. I am having trouble moving forward from this information and how to construct a proof out of all of these.
Thank you.