A linear normed space as the union of open convex subset

Question

Let $K$ be an open convex subset of a normed linear space $X$ over $R$ and $0 \in K$. I want to prove that $X=\cup_{t\geq 0} tK$. I know how to prove $\cup_{t\geq 0} tK\subset X$. However, I'm not sure how to prove the reverse, i.e., $X\subset\cup_{t\geq 0} tK $. Can someone help please?

Answer 1

Since $0\in K$ and $K$ is open, we have some $\delta>0$, $B(0,\delta)\subseteq K$. For any $x$, choose $\epsilon>0$ small enough such that $\epsilon\|x\|<\delta$, then $\epsilon x\in K$ and hence $x\in\epsilon^{-1}K\subseteq\displaystyle\bigcup_{t>0} tK$.