Rudin Functional Analysis regarding the Minkowski functional

Question

Let $A$ be a convex absorbing set on some vector space $X$ and $x\in X$. In the proof of theorem 1.35 of Rudin's Functional Analysis, he states that if $\epsilon\in(0,\infty)$, then $t:=\mu_A(x)+\epsilon$ is such that $t^{-1}x\in A$. However, I don't see why this is true. It seems to me that he is using something of the sort: if $t>\mu_A(x):=\inf\{\lambda\in(0,\infty)|\lambda^{-1}x\in A\}$ then $t^{-1}x\in A$. However, the condition $t>\mu_A(x)$ only guarantees that $t$ is not an upper bound, i.e. that there is a positive $\lambda<t$ such that $\lambda^{-1}x\in A$. Of course, if $A$ was assumed to be balanced, one would then have $t^{-1}x=t^{-1}\lambda\lambda^{-1}x\in t^{-1}\lambda A\subseteq A$, given that $t^{-1}\lambda<1$. This however was not assumed. Is there something obvious I am not seeing?