Closed convex absorbing set has an origin neighbourhood
= Closed convex absorbing set has an origin neighbourhood
Let $S$ be a nonempty closed, convex, symmetric subset of a Banach space and suppose $\bigcup_{n\geq1}nS=X$. Baire's theorem puts a ball $B(x,r)$ inside some $nS$. Symmetry also puts $B(-x,r)$ there, and convexity puts the midpoint of $x+z$ and $-x+z$, namely $z$, in $nS$ whenever $\|z\|<r$. Hence $B(0,r/n)\subset S$.