I usually have a lot of trouble with complex variable when it comes to the geometric representation of $\hat{\mathbb C}$ and what happens in there. I have the next exercise and from quick look at it I don't think is difficult to prove, but since I don't understand very well the Riemann sphere I miss a lot of things.
- If $a\in \mathbb C$ and $r>0$, then there exists $\rho>0$ such that $B_\infty(a,\rho)\subset B(a,r)$.
- If $\rho>0$ and $a\in\mathbb C$, then there exists $r>0$ such that $B(a,r)\subset B_\infty(a,\rho)$.
- If $\rho>0$, then there exists a compact $K\subset \mathbb C$ such that $\hat{\mathbb C}\setminus K\subset B_\infty(\infty,\rho)$
- If $K\subset\mathbb C$ is compact, then there exists $\rho>0$ such that $B_\infty(\infty,\rho)\subset\hat{\mathbb C}\setminus K$
For the first one. Before anything I tried to see what $B_\infty(a,\rho)$ looked like so I would relate it to the usual $B(a,r)$. $$B_\infty(a,\rho)=\left\{z\in\hat{\mathbb C}\,:\, d_\infty(a,z):=\frac{2|a-z|}{\sqrt{(1+|a|^2)(1+|z|^2)}}<\rho\right\}$$ Now from what I understand, the metric in $\hat{\mathbb C}$ measures the distance bettween the projections in the sphere of two points in the plane $\mathbb C$ or maybe I'm getting it backwards or just wrong. So the ball $B_\infty(a,\rho)$ would be the projection on the sphere of the points in $\mathbb C$ such that satisfy $d_\infty(a,z)<\rho$, so in my imagination I think it looks like an open disk on the sphere or maybe like an elipse. I already tried getting from $B(a,r)$ to this $B_\infty(a,\rho)$ with algebra, by taking a $z\in\mathbb C$ such that $|a-z|<r$ and then I tried to build $\rho$ to get $d_\infty(a,z)<\rho(r)$, but I can't anywhere specially since the geometrical interpretation is not clear for me.