Convex subsets of a topological vector space

Question

I'm trying to prove:

Let $X$ be a topological vector space and $A \subseteq X$. $A$ is convex if and only if $$\forall s,t \in \mathbb{R}_{+}, (s+t)A = sA + tA $$

Let $sx + ty \in sA + tA$.

How can I use the properties of convex spaces, which use only real numbers in $[0,1]$, on such a element of $A$?

Answer 1

Hint: ${s\over{s+t}}x+{t\over{s+t}}y\in A$ if $x,y\in A$ and $A$ is convex, thus $(s+t)({s\over{s+t}}x+{t\over{s+t}}y)=sx+ty\in (s+t)A$