# Calculating the convex hull of a set.

I have a question about a answer in this post: convex hull specifically from the answer given by @Per Manne.

I know that $$B=\left\{ \begin{pmatrix} x \\ y \end{pmatrix}\in \mathbb{R}^{2}: y>0\right\}$$ is convex set, I could easily verify using the definition of convexity (in fact geometrically it is very intuitive), but how can I prove that $$B$$ is convex hull of $$A=\left\{\begin{pmatrix} x \\ y \end{pmatrix} \in \mathbb{R}^{2}: y \geq \frac{1}{1+x^{2}}\right\}?$$

I think that intuitively it is true, since it is known that by extensivity $$A \subseteq conv(A):=B$$.

But, by definition how can I prove that fact?

Let $$(x,y) \in B$$. Let $$n$$ be a positive integer such that $$n >|x|$$ and $$1+n^{2} >\frac 1 y$$. Let $$t=\frac {n-x} {2n}$$. Then $$0. Now $$(x,y)=t(-n,y)+(1-t)(n,y)$$. Note that $$(-n,y)$$ and $$(n,y)$$ both belong to $$A$$. Hence any element of $$B$$ is a convex combination of two points of $$A$$.