obviously for $a,b \in ]\frac{-\pi}{2},\frac{\pi}{2}[$ the ordinary tangent map is strictly monotonic. Hence for $b \geq a \Rightarrow \tan(b) \geq \tan(a)$. In the proof I try to understand it follows that $\tan(]a,b[)=]\tan(a),\tan(b)[$. I need this since once proven I can conclude that tan maps open sets to open sets. It is clear that this statement is indeed true, but how to prove?
Thanks for your help!
$$x\in (a,b)\Longleftrightarrow a<x<b$$ $$\Longleftrightarrow a<x\ and\ x<b$$ $$\Longleftrightarrow tan(a)<tan(x)\ and\ tan(x)<tan(b)$$ which follows due to monotonicity of tangent function $$\Longleftrightarrow tan(x)\in(tan(a),tan(b))$$
Hope it is helpful