Showing that $S$ is not left amenable if it has two or more left zeros

Question

I'm studying analysis on semigroups by myself.

Let $S$ be a semigroup. Show that if $S$ has two or more left zeros then $S$ is not left amenable.

For proof, let $\mu\in \operatorname{LIM}(S)$, then for every $f\in B(S)$ and $s\in S$ we have $\mu(L_sf)=\mu(f)$. I stop here and cannot find a contradiction. Please help me.

Answer 1

I keep your notation. Suppose $s$ and $t$ are two left zeros. Then $\mu(L_s f) = \mu(L_t f)$. However, for all $x\in S$, $(L_s f)(x) = f(s)$ and $(L_t f)(x) = f(t)$. Since this holds for all $f$, it follows $s=t$.